How to politely pursuade a lay person/Bhante to wear a mask?

One isn't either forced nor asked to seek refuge in the Gems, yet monks would offend if teaching to those who have no respect and if one's wearing a cover that would be a outer sign of no respect and actually not seeking refuge, still to little and RED (sign of greed, anger, strong attachments to views and world: marxist/materialist) and yes, actually not proper to wear things similar like householder.

One is free to seek refuge elsewhere and even avoid the Gems.

Yet, of course, aging, sickness, death can not be overcome by else then the Noble Path. Gross the pandemic of wrong view, ingratitude, demand....

Well, all you can do, really, is remind the person in question that it is against Buddhist precepts to take a life, and that this is as true of acts of omission as it is of acts of violence. At the very least, it will open an interesting philosophical debate. Monastics are likely not overly concerned with the state of the physical body, but people who attend then are still caught in that ignorance. Monastics ought to respect that.

Carrying extra masks is a fine, compassionate idea. Some will take them, some won't. Those who don't have a desire they are not willing to address, and may not even understand, so show them compassion.

If someone refuses to wear a mask, they refuse. Remember that the goal of buddhists isn't to put an end to disease; the goal is to put an end to ignorance and its karmic manifestations. Don't allow their refusal to become grounds for increasing ignorance; don't allow it to disturb your own equanimity. They are fighting against something illusory, and you don't want to get drawn into that world.

Nobody forces one to be next someone or something one doesn't like. So what's the real problem? It's to assume that the monk wouldn't run after good little RED, and his reasons are his. It seems that good householder forgets that demanding is always caused by wrong view, lacks of metta and path as whole. Right?

The monastic rules of the Vinaya has rules pertaining to food, lodging, medicine, conduct etc. There are rules pertaining to medical care. The Buddha ate, slept, wore robes, took medicine when he was ill etc.

The Middle Way of the Buddha avoids both extreme indulgence and extreme asceticism. It includes moderation in food, healthcare and living. To neglect healthcare is to go against the Middle Way of moderation.

From Kucchivikara-vatthu of the Vinaya, the Buddha taught:

"A sick person endowed with five qualities is hard to tend to: he does what is not amenable to his cure; he does not know the proper amount in things amenable to his cure; he does not take his medicine; he does not tell his symptoms, as they actually are present, to the nurse desiring his welfare, saying that they are worse when they are worse, improving when they are improving, or remaining the same when they are remaining the same; and he is not the type who can endure bodily feelings that are painful, fierce, sharp, wracking, repellent, disagreeable, life-threatening. A sick person endowed with these five qualities is hard to tend to.

"A sick person endowed with five qualities is easy to tend to: he does what is amenable to his cure; he knows the proper amount in things amenable to his cure; he takes his medicine; he tells his symptoms, as they actually are present, to the nurse desiring his welfare, saying that they are worse when they are worse, improving when they are improving, or remaining the same when they are remaining the same; and he is the type who can endure bodily feelings that are painful, fierce, sharp, wracking, repellent, disagreeable, life-threatening. A sick person endowed with these five qualities is easy to tend to.

Based on this Vinaya quote, I would say that the Buddha himself would have wanted his followers to practice moderation, and accept the advice of healthcare professionals and the government when dealing with the Covid-19 pandemic.

Also, spreading misinformation about Covid-19, masks and vaccines is against the fourth precept of not speaking falsehood.

Wearing a surgical mask is more for the wearer to avoid spreading pathogens to others, than for the wearer's own protection from contracting an infectious disease. As such, wearing a mask with the explicit intention of reducing others' suffering, is also an expression of compassion (karuna).

The quote from DN 26 below, also shows that the Buddha did not promote anarchy. He stated that the wheel-turning monarch should provide just protection and security to his subjects and also care for their welfare.

‘But sire, what are the noble duties of a wheel-turning monarch?’

‘Well then, my dear, relying only on principle—honoring, respecting, and venerating principle, having principle as your flag, banner, and authority — provide just protection and security for your court, troops, aristocrats, vassals, brahmins and householders, people of town and country, ascetics and brahmins, beasts and birds. Do not let injustice prevail in the realm. Pay money to the penniless in the realm.

I would even say that a responsible Buddhist should support his or her government and follow its advice, if the government is discernibly doing the right thing, and if its advice is in accordance with the five precepts, Right Action, Right Speech and Right Livelihood. For e.g. conscription into military service is against Right Livelihood.

You can use these arguments to politely persuade monks and lay Buddhists.

Once upon a time, monks, a bamboo acrobat, setting himself upon his bamboo pole, addressed his assistant Medakathalika: "Come you, my dear Medakathalika, and climbing up the bamboo pole, stand upon my shoulders." "Okay, master" the assistant Medakathalika replied to the bamboo acrobat; and climbing up the bamboo pole she stood on the master's shoulders.

So then the bamboo acrobat said this to his assistant Medakathalika: "You look after me, my dear Medakathalika, and I'll look after you. Thus with us looking after one another, guarding one another, we'll show off our craft, receive some payment, and safely climb down the bamboo pole."

This being said, the assistant Medakathalika said this to the bamboo acrobat: "That will not do at all, master! You look after yourself, master, and I will look after myself. Thus with each of us looking after ourselves, guarding ourselves, we'll show off our craft, receive some payment, and safely climb down from the bamboo pole. That's the right way to do it!"

This is the teaching of the Buddha. You wear your mask, and let the others take care of theirs. No need to fix them.