I live in a country with a relatively similar political climate as Poland (highly religious, post-communist, wannabe central Europe). And I used to use the same argument when I was surrounded by more conservative people. The argument is IMO frequently invoked not by people who are truly worried about children (which I'll write about below), but by conservatives who need a civilised, "agnostic" argument for their homophobic stances. But ofc it's better to assume good intentions, at least if you don't know anything about the person using the argument (as e.g. here).

The biggest problem with the argument is that it's purely reactive and, under the hood, disingenuous. Children bully each other horribly already for a million stupid reasons - their shoe brand, their phone brand, their behaviour, etc. or just so, for no detectable reason at all. They also bully their teachers and professors. What is done against all this? Absolutely nothing, as far as I see (and I've seen and heard plenty while I was growing up). It is never brought up as a problem in public discourse, nobody seems to care too much. Bullying somehow becomes a big problem and relevant for the lawmaking only when gay parents are a possibility.

In general, from what I've seen, bullies will find just about any reason to target a kid. Adding one more to the roster seems borderline trivial. E.g. a lot of existing bullying is class-based - my younger sister was mildly ostracised in the primary school for a while because she wore the clothes my mother sewed for her, without a brand or anything, suggesting we don't have the money to buy "proper" clothes. Should we, then, try to separate poor kids from the rich kids, so the poor don't get bullied? Or just forbid poor kids from going to school?

Thus, instead of doing anything against the actual problem – that is, bullying as such – the laws of the state, the fundamental right of a child to a family, etc. should all buckle down before some child bullying? A child should be denied growing up with a potentially good and loving family with LGBT parents, and instead be adopted by a potentially inferior heterosexual family (assuming the adoption centres have some sort of system to judge the adopters in advance), or stay without a family at all indefinitely, because someone could/will bully them based on their most intimate and safe space, that is their family? Just as it would be monstrous to forbid poor kids from going to school to "protect" them from bullying, it is monstrous to propose "to protect some kids from bullying, we'll deny them from having a family". The whole argument is actually (or should be) an argument for aggressively rethinking and reworking your educational system , parenting and culture in general.

because why should these children be victims of war that is not even theirs to fight

Under the current system they're also victims and involved in this same war - a part of their potential adopters is denied by default, and they stay without a family for longer. Are they not victims here? (Not to get into the issue of measuring potential benefits of having a family against the potential negatives of bullying, it's purely arbitrary and depends on the given culture too.)

On the other hand, I do think the whole discussion has been derailed by overly focusing on this as an LGBT issue rather than an issue of children without families. So there's some merit at least in the general approach of the argument you present (the children are those whose well-being is most important here), but it leads to the wrong conclusion, usually because it's invoked by people who really just want to get to that conclusion one way or another, rather than helping the kids.