Let me start by emphasizing this is purely speculation, but to give an example of "GMaxwell's motivation [to] be for trying to benefit BFL in any way":
a. BFL has spend a substantial amount of advertising on BCT;
b. BFL has laundered millions through Bitpay;
c. once the "BFL fucked us over again" topic started finding more and more evidence of that this "lawyerdude" shows up intentionally derailing the thread;
d. dude gets doxed;
e. GMaxwell all of a sudden (yes, sudden) starts moderating the bejeezers out of cet topic...
Considering the amount of doxxing previously going on by BFL representatives (althoug you seem to find that insignificant) this seems highly suspitious.
So maybe BTC in general, or GMaxwell in particular still have something to gain from either BFL or Bitpay
Again: SPECULATION, but motivation enough for you?
Well, I can answer a few of those for you, others I don't know.
A. You are right, BFL has spent a substantial amount of advertising on BCT, not in the last 6+ months, but you are correct. Back when they were advertising and this came up very frequently, mainly about corruption of Theymos, I did a calculation. BFL's ad purchases equaled an additional $1.17 per month paid to a moderator. So pretty good indicator that no moderators were into BFL for the money.
B. No idea if GMaxwell has any connections to Bitpay, I wouldn't necessarily call that a very plausible thought, but its not outside of the realm of reality I suppose.
C. Lawyerdude intentionally derailing the thread, doesn't that give reason to delete his posts?
D. Moderators do not remove Dox. The thinking behind this is that Dox tend to be public information. If you can obtain it without breaking into somewhere to get the information, its public info, ie phone numbers, name, addresses, etc all stuff you can find with a google search or in a phone book if those still exist.
E. I'm not sure what his motivations are, but I've seen it more than a few times. A thread will start out with a controvesial topic, proponents and opponents clash, people that are bored get involved for giggles, a web of spam and insults occurs, it gets exponentially worse to the point where the only way to handle it is a clean sweep through the entire thread.
That said, besides being a moderator GMaxwell is a Bitcoin core developer, BFL hasn't been the best shining example of why people should get involved in Bitcoin, so as far as I can still tell there isn't any reason for Gmaxwell to want to help BFL. That is unless he has some connection to Bitpay as you said as a possible option, which I don't think he does, but I do give you points for writing out feasible reasons, rather than that GMaxwell is actually an alien that probed Josh once, and liked what he found, so he now has a foundness for BFL or something like that.