Could dhammanupassana be to remember satipatthana?

Could the 4th satipatthana or dhammanupassana be, among other things, to remember Satipatthana in both heart and mind?
Main purpose of various vipassana is to rectify defiled behaviour pattern.

to remember Satipatthana in both heart and mind
Is it referring to passing of meditation-knowledge to next-birth through Citta? Reference
If so, then be assured that it's the behaviour pattern that travels to next birth, otherwise buddha would have taught only satipatthana to everyone whereas, he taught only to rectify defiled behaviour pattern through different Upadesh(lessons cum teachings) to different people.

To think again, What would it(remembering) benefit Citta, if defiled behaviour pattern(within) cannot be rectified?
Nothing

Shouldn't viriya be focused on the focus towards satipatthana when worldly matters have distracted one down a wrong turn on the path?
----:) focused on focus towards satipatthana, where satipatthana in itself is not enlightenment rather a path to enlightenment. Seems like a similar idea used in hindu geeta by krishna for setting-up arjuna-- "I am teaching you a path upon walking on which, you will reach a path that will ultimately reach to me".

Other than stream-enterer, everyone is on wrong path. Stream-enterer is on a path that will reach to enlightening. Others are focusing on the path to attain stream-entry.

If it's about meditation procedure, then from practical experience::

sorry to say, in long-run meditation this focused part won't survive. Meditator will ultimately have to switch to, "focus on sattipatthana' part, still in further journey this focus also won't survive, as meditator will have to switch to, "realizing defilements and rectifying behaviour".

Theoritically, this realizing also won't survive in further running, meditator will have to switch to, "stop worrying/analysing about behaviour" and achieving the state of "impermanence at atomic or tiniest scale" after which there is nothing left to achieve as achiever itself is 'causes&conditions, an impermanence', no-where left to run.

Do anything Buddhist scriptures talk about this?
Again, to focused towards focus on satipatthana part, many teachings in suttas are there for right livelihood through viriya.
One of the reference being Reference 2

The meaning of dhamme-dhamma anu-passi viharati, means, To see/contemplate Dharma as the Dharma [actually is]. A similar phrase in the suttas that occurs frequently is Dhamma-anu-Dhamma patipada, practicing Dharma in accordance to Dharma. Some passages explain that here: http://notesonthedhamma.blogspot.com/2019/05/dhamma-anu-dhamma-practicing-dhamma-in.html

Dhamma/Dharma primarily means the natural laws and teachings of the Buddha that lead one to Nirvana. In other words, it's not just any 'phenomena' or 'mental quality' you're contemplating, or should be contemplating. You should always be practicing for the Dhamma that leads to Nirvana, not dhamma/phenomena of ultimately pointless things.

Yes. Satipatthana (Samma Sati) is an important path factor in Noble Eightfold Path. Viriya or Samma Vayama come before Samma Sati (Satipathana). Samma Vayama, Samma Sati and Samma Samadhi all three come under Samadhi (concentration) category.

Your question is about the 'goal' of dhammanupassana. You are rightly careful to not think of it as the only 'benefit' of dhammanupassana.

Establishment of sati, is fourfold. Fourfold establishment means that sati is perfected when sati (awareness) is established over the 4 categorized mind-body phenomenon. That is to say, mind-body phenomenon is viewed in terms of the 4 categories, awareness of body (kayanupassana), awareness of sensations (vedanupassana), awarenesss of mind (cittanupassana) and awareness of the nature/law of mental-contents (dhammanupassana). Pefrection of sati is gradual and happens in all 4 categories (since there are no 'categories' as such, but the term is used conventionally to make sense of mind-body phenomenon). Of course, establishing awareness of body is easier than the rest, nonetheless, perfection (perfect establishing of sati) which is to say , complete awareness happens only when progress is made on all '4' categories.

I do no unedrstand what you mean by heart in this context. Memory is a mental phenomenon and so perfection of dhammanupassana, of course, results in sharper and clearer memory (only as an off-shoot; not the main goal(or even an intended goal) of dhammanupassana). As to the question about 'viriya', while in a subtle (and indeed profound) sense, viriya specifically implies for the meditation period, in a superficial sense (very usefull and certainly not wrong) it also implies to right effort in one's everyday life.

But to answer your question in the simplest way, in the context of meditation, certainly viriya must be applied to get back on the path as one strays and becomes aware that one has strayed. Just dont stop with viriya there!

References- You may refer to Sattipathana sutta and Mahasattipathana sutta for better understanding based on your own reading.

NOTE-

1. I have used 'awareness' as the english translation for sati. 'Mind-fullness' works equally well in this case.
2. I have used 'effort' as the english translation for viriya. 'Energy' would be an equally goood translation in this case.