My current file is working with Yes Man, and I am to the point where I will be making allies. My first choice for a group to go to was randomly picked as the BoS. Unfortunately, each time I get near the place that Yes Man thinks they are hiding out, a group of BoS Paladins attack me with miniguns, gatling lasers, and fat men/mans. They wipe the floor with me pretty easy, since I'm charging in with metal armor and a cowboy repeater. So, the question is, how do you approach the Brotherhood without pissing them off?

Have you tried going with Veronica in your party?

I'm not sure where to find Veronica

She's at the 188 trading post. Look for a woman in brown robes weilding a power fist.

Well for me, for one of my "saves," I was trying to get on better terms with the Brotherhood so they didn't have to die. In order to do that, eventually there's a quest for the NCR to go convince the Elder to join NCR to stop the violence. However, I did all the other Brotherhood quests (especially the one where you try to get the Elder overthrown). That prevents you from being to able to get the BoS to join up with NCR.

In my opinion, if you want the Brotherhood to live, you kind of need to wait to do their quests until towards the end.

Or, you just don't do the quest to overthrow the elder- the elder has his own alternate quests for you instead that will still lead to you joining the BOS, getting PA training, etc. And with the old elder convinced to remove the lockdown, he can also be convinced to let the BOS join the NCR.

The new elder is actually less progressive than the old one and wants to attack helios one immediately.
As for you OP, I would recommend going to 188 and getting veronica, then take her to McCarran, the Followers, the Silver Rush, etc and then she'll want to go to the BOS bunker. Then they shouldn't be hostile to you.

You can just tell Yes Man to ignore the BOS.

I took Veronica with me and it worked like a charm. Let me into the bunker, and the patrols weren't hostile to me.