I'll give a counter thought, with a giant caveat of "it depends"... Individual nuances will vary, so take this FWIW.

Bid equipment first, then bid to change bases. This is assuming you would like equipment that is in nearly every base but the one you want (i.e. 320/73N, and to a lesser extent 7ER), or your SSN isn't quite high enough to get the base you want on the initial drop but high enough to get the equipment you want. We traditionally have about 5 Advanced Entitlement (AE) bids annually - it constantly varies. You can always bid immediately to change bases on the same equipment - because there is no seat lock to the base, just the equipment. It's not a guarantee, but I submit it's a low risk/high reward strategy in most cases. You might have to commute for a couple of months, but you also might not ever have to report to your original assigned base. This happened to me on the -88 when I got hired into NYC. I got lucky with AE timing, but was converted to ATL before I was even done with OE. Winning.

Every AE is different, so you have to be careful extrapolating trends from just one AE, but... It seems most AE's have seen nearly every NH category with super junior folks switching bases. Remember, every category in a NH drop can ONLY be from positions left unfilled from the most recent AE. Some bases that have been traditionally more senior (looking at you, MSP) seem to be trending more junior as the NW pilots retire - this may be the only way to get to MSP (which rarely show up in a NH drop).

Having said that, if your family situation wouldn't tolerate that risk, go for the base over equipment, then wait out the 2-year seat lock and switch equipment. But if you want to avoid a month on VA Ave twice, there is a decent chance to be successful by bidding the equipment, then switching bases. Good luck.