What are the consequences of a monk eating meat?

A Monk can't be a strict Vegetarian

If a monk follows real Buddhism, he can't be a strict vegetarian and refuse what has given. Many vegans see a living animal when they see meat. One who follows the path of Buddha doesn't see a living animal when they see meat . So they don't know/see a difference between a vegetable and a meat . Just some food in the bowl to keep the life going. If they identify food in their bowl as , this is delicious, this is not delicious, this is expensive, this is cheap , this is meat , this is vegetable, that's not real practice. And if they see a living animal in a dead meat, that means they take "Ruupa" (what we see through eyes ) as a permanent thing . The impermanence of "Ruupa" is what we/monks should practice to see . Thus a monk can't be a strict vegetarian if they practice what the Buddha has taught .

What one grasp with his hand from his bowl and puts into ones mouth is the last in line of ones decisions. If one thinks he needs to act discriminatory there, there is all, possible lesser other annoying possibility there. That should satisfy good householders concerns for going on.

My answer is based on the Chinese Classical Sutras I read, there are three situations:

A) A Bhikkhu should NOT eat meat because he is on the path of liberation cultivating the Caturapramāṇāḥ/四無量心 (Four All-encompassing Hearts), due to Maitr/Compassion he should not kill so eating meat is participating indirectly or encouraging killing.

B) A Bhikkhu should practice equanimity he should accept whatever he offered. If in his meal there is meat, he should wash it with water until it has no taste, eat it as if eating the meat cut from his own son in the time of famine. That is the Buddha's true teaching.

C) If the offer is all meat or mainly meat. A Bhikkhu should choose not to accept.

The main reason for obstaining from eating meat is it hindering cultivating Maitr/compassion, meat eating promotes craving of taste. To a subtle level, meat eating the smell of meat left after eating attracts Rākṣasa/羅刹 those blood-flesh thirsting Monsters; sending away the Celestial Beings those who are repellent to meat eating, Suangama Sutra. The Celestial Beings seeing human eating meat the repellent is the same as human seeing dogs eating feces (though modern day dogs eating beef or canned dog food different from the pre-modern time).

The mentioning of Celestial Beings and Monsters for modern day Buddhism is like terms from sci-fictions, but even the Theravadist modernized Pali Canon still maintains verses about the Bhikkhus encountering Beings of higher realms at this moment, I read from the quotes of this forum. A Bhikkhu with correct practice should be able to open his perceiving faculties to sense, if not seeing, realms different from the human and animal's. A Bhikkhu once on The Path he will be guarded by the Beings those vowed to the Buddha to uphold his Dharma these guarding Beings may leave him when repelt by the smell of his meat eating. However, those who have the robes on are not necessarily a Bhikkhu on The Path there were no guarding Beings around him either. It's like there are many discussions or even debate on Anatta, we write, speak, even graffiti-ing this word but we don't really know, not to say have a taste of, Anatta.

Last, there a situation a Bhikkhu is allowed by the Buddha to eat meat, when he is practicing Dhyana of a practicular type related to White Skeleton Visualization of the Mahayana (Chinese) method, else he will be going mad. In this particular situation he must eat meat but when he does, he has to do B), wash the meat to no taste, eat it as if eating meat from his own son.

These are the teachings about meat eating addressed solely to the Bhikkhus, not laity.

since the Buddha didn't prohibit monks from eating meat

Therefore this is of dispute, it should be only the modern Theravadist understanding, or maybe with some schools like the Tibetan Buddhism. By what I studied in the Chinese Classical Sutras the principle is Buddha taught Bhikkhus NOT to eat meat, but with flexibility to accommodate the different circumstances. Buddha himself didn't eat meat, even the Brahmins of modern day India don't - I personally has an acquaintance. For Tibetans, Buddha had once addressed certain Brahmins at that time due to their geological condition crops didn't grow there they were allowed but with certain conditioning to eat, that may apply, or do it as B).

how should/would a monk who has chosen to be a vegetarian handle being offered meat by a lay person?

Therefore, a Bhikkhus has chosen to be a vegetarian is, from the principle about meat eating taught by the Buddha, also an unwholesome or unnecessary practice. If this for whatever reason he has to keep to be a vegetarian he may do B) + C). Or, consider the Chinese Sangha. The Chinese Sangha, due to the historical cultural background, they evolve to a Farming-Ch'an/農禪 style practice they farmed their own food instead of going for alms everyday, farming labouring is a form of Dhyana. Sometimes in times of war or famine they may even offer shelters or food to the people, in accord with Mahayana's Bodhisattva vow. Recently there is also a monastery reviving the practice of alms going.

Buddha didn't prohibit monks from eating meat

Yes.

monks are supposed to be easy to maintain and take what is offered

Yes.

how should/would a monk who has chosen to be a vegetarian handle being offered meat by a lay person?

A monk shouldn't choose. The monk should accept the food (in his alms bowl) and can choose to not eat the non-vegetarian part later. If a monk always uses his bowl, including when invited to a meal in a home, then the offerers cannot see what he eats. The monk can pick & choose in his bowl, although there will be meat substances polluting his vegetarian choice.

If they chose to refuse the meal would that make them high maintenance, proud or unappreciative?

They should not refuse. To refuse would be extremely evil & Satanic.

If they ate it, how would they reconcile that with their personal practice?

If the monks is practising Buddhism (rather than Hinduism), eating meat should not affect their practise since the practise of Buddhism is non-attachment, including towards food.

In summary, eating meat is not a sin that needs to be reconciled. Instead, any idea of 'reconciliation' is a personal delusion. Attachment to vegetarianism is a grave & fatal fetter because it most likely prohibits reaching non-returner & arahantship.

I'm particularly interested in an answer from any monastic on how they have or would handle this.

I have lived in a remote monastery before with forest monks. All of the alms food, including the fruit & curries, were emptied into a bowl, mixed up, and then spooned out individually.

A real monks does not pick & choose alms food. That is a disgrace.

You can find an answer from a Theravada monk, here:

Despite a widespread belief to the contrary, the Vinaya doesn't actually oblige a bhikkhu to ever accept any offering from anybody. And so in theory a bhikkhu could walk on almsround ignoring all the requests to stop from donors who offer food that he doesn't like. In so doing he wouldn't be breaking any Vinaya rule. Socially, however, this would be quite a serious faux pas in Asian Buddhist countries. As a bhikkhu one is expected to be encouraging and facilitating dāna on the part of householders, not obstructing it, and so a bhikkhu's rejection of a householder's allowable offering would be viewed as perverse and mean-spirited.

If he's living alone and has a sensitive digestive system, he should accept whatever he's offered but eat only what agrees with him. If he's living in a monastery, then he can make arrangements with his fellow monks after notifying them of his condition. In either case, if he's seriously ill then the Vinaya allows him to request special foods.