NC'd for this but am a regular. This is going to be a bit of an essay, so apologies in advance.
Long story short, DH and his DB fell out about 4 years ago. This was DH's (then DP's) fault - they lived together at the time, I'd not long escaped a violent relationship (DH and I got together soon after, we'd been friends previously though), and DH moved me into his house with my baby DS. It was a big move for me, about 5 hours from my hometown. DH told me he'd cleared it with his DB, and that he was fine with me moving in - there's no way I'd have done it had he not been okay with the idea.
It turns out that no, his DB was not okay with it. He confronted me when we were on our own one day, and said he'd had no idea I was moving in, and was seriously pissed off about it. I was shocked, told him I was under the impression that he'd known and given his approval, and suggested we talk to DH together and try to sort it out. He stormed off. DH came home from work, I asked him what the fuck was going on, he admitted that he'd only sent his DB a text to tell him I was moving in and had had no reply! Now, I KNOW DH is the one who fucked up here, and there's no denying he was absolutely in the wrong.
The atmosphere at home was awful. BIL ignored us both, unless I was in the kitchen washing up, in which case he'd dump all his dirty plates in the sink and storm off. Eventually, I'd had enough of it, and told DH to clear the air with him. DH has always seemed a bit scared of his (10 years younger) DB, and was reluctant too cowardly to 'upset' him, so I knocked on BIL's door and told him this needed to be sorted out once and for all, as the current situation was helping nobody.
BIL told me to fuck off. I repeated that we had to sort things out. He shoved me, hard, knocking me over. He then proceeded to slam my foot in the door several times, resulting in a trip to hospital to check nothing had been broken. DH helped me up, but said nothing to his DB. As I mentioned above, I wasn't long out of a violent relationship, and I was distraught.
BIL was forced by MIL to apologise to me, but it was completely insincere and wouldn't have happened had MIL not put him under duress. The atmosphere continued til BIL moved out a few months later. He refused to speak to DH following that.
DH's DGM died recently, and her funeral was a couple of weeks ago. I had the DC so couldn't attend, but DH went on his own. BIL was there, he shook DH's hand and they made small talk. DH seemed disproportionately delighted about this - in the past, he'd had very low self esteem, and always tried to please the wrong people. The worse he was treated, the more he'd try to do to please that person. That might have some bearing here.
On his return home, DH sent BIL a text, thanking him for shaking his hand 
and saying he hoped they could move on. BIL hasn't replied, and it doesn't look like he had any intention of doing so - no doubt he was keeping the peace in order to save my already distraught MIL from any further upset at the funeral, but had no real interest in smoothing things over with DH.
Now DH is very eager to build bridges, and he's upset that I'm not very supportive about it. Don't get me wrong, I completely understand that falling out with close family is awful, but I know DH is very often too keen to sweep past issues under the carpet. He's grown a lot of backbone in the nearly 5 years we've been together, and doesn't take nearly as much shit, but this business with BIL is a bone of contention between us. The thanking him for the handshake just came across as desperate to me, like a dog who still rolls over even when it's been kicked. DH says I need to stop bearing a grudge, and to support him, to which I replied how can he be so happy to make up with someone who assaulted his now wife?
I really don't know what to do for the best here. I'm the first to admit that I'm a champion grudge holder - I have BPD, so I'm all too aware that my thinking is very black and white when I've been wronged - but I don't think that BIL should be forgiven so easily. I know for a fact that DH wouldn't ever bring up what happened, nor would he say that I needed a proper apology. I've made it clear that whilst DH is free to build bridges with BIL, he's not welcome in my house (and yeah, I do get the irony there!).
So what do I do? Do I let this go for DH's sake? Is there any way around this? Thanks for reading, if you got this far!