We can't debate this - it breaks up families

Foundationally, this is an opinion. If we do X, the results will be Y, and Y is too steep of a price to pay for X. While one can appeal to emotion in this plea, it wouldn't be a fallacious appeal, as the appeal would be referring to the act of debating itself, not the claim. In other words, the appeal to emotion is not an attempt to convince someone that the claim is right or wrong; it is to convince someone that the act of debating comes with too high of an emotional cost. I think most people can sympathize with this argument who try to stay away from ideologically-charged social media posts, knowing the consequences would be too great (e.g., spending the better part of your day defending your position).

I don't know about what fallacies there are hidden in that argument, I think you covered them. I just want to point out that we are experiencing the same problem here in the US. We have thousands, if not millions of families who have been dismantled recently. But yours is based on a massive ideological debate. Ours is based on an idiotic game show host.

'We shouldn't argue about this. It breaks families up.' 

Logical form:

We should not argue about things that break families up.

This issue will break up the family.

Therefore we should not argue about this issue.

If the major premise is true, then the conclusion does in fact follow, so the argument is formally valid. However, the premises are unacceptable because they rely on assumptions that aren't logical themselves. First, we see the Appeal to Emotion as the speaker is begging us not to discuss something - even if it is important. The stated reason is 'we should not argue about things that break families up' which conjures emotionally unpleasant imagery, hence we abandon reason and the conversation. There is no consideration of the relevant urgency of the issues; anything that 'breaks families up' is off the table. Secondly, there is no guarantee that this issue will break up the family; for instance, if the debate is cordial and controlled, animosities may be resolved or minimised.

'So you want to break the family up, is that it?'

This is a very blatant Strawman Fallacy, as you correctly identify.

Instead of intelligent, coherent discourse, defenders of tradition and blind loyalty use this tactic to avoid defending their otherwise untenable and weak positions.

Yes, this is a very common occurrence in any sort of contentious issue. Rather than accepting a challenge to their long-held beliefs, they would rather beg you to not think about the many ways they could be wrong.

There's no specific fallacy I can think of, but the ones I've given above will suffice. Avoiding the Issue could also apply here, if the speaker strays off topic to avoid making an argument.