I can't help with your battery question I'm afraid, but I do have the B&G wireless wind instruments so may be able to offer a few comments on that, and can comment on the relative merits versus the Garmin gWind unit.
Like you I also have the B&G Vulcan plotter, and wanted to be all B&G from the outset, but in 2016 B&G didn't have a wireless wind offering, so I went with the Garmin version. Interfacing into the N2K network was mildly complicated needing two interface boxes and a laptop connection to set up the software, but it went together ok. What I found after not that long in use, perhaps 6 months, was that the signal kept dropping out for short periods. It was like it was feeling in need of a rest so it went off for a fag and cup of tea for 15 minutes, but it always came back again. That is until mid 2019 when it went off and never came back. I contacted Garmin about getting it repaired and they said it couldn't be fixed as they were now making the Mk 2 version and didn't have any Mk 1 PCBs left! This is on a 3 year old product, so not impressed. They offered a Mk 2 upgrade at reduced price, but the Mk 2 now only interfaces to NMEA 2000 via one of their plotters or readout units so no use to me. They also recommended that the wired version would be a much better option, even than the Mk 2, so not much of a vote of confidence from the manufacturers then.
By this time B&G had brought out their WS320 wireless unit, which they claimed had done 20,000 hours of testing, so sounded like a more reliable option, so I decided to switch to that. It's about the same size as the Garmin, but seems to me to be rather better designed and made. On the Garmin the vane unit was not very well balanced, so if you roll it over sideways the impeller swings round until it's vertical. This is not helped by the fact that the speed impeller and the direction vanes are both on the same pivoting unit, and when you assemble the impeller onto the spindle then the balance is affected by how far you push the impeller on before tightening the grub screw, so a bit fiddly to get right. On the B&G the speed impeller and direction vane are separate and pre-assembled. The balance is absolutely perfect and the bearings are very low friction, so you can tilt and rotate the support wand around underneath it and the vane stays perfectly still. I've also noticed that in very light winds the B&G anemometer cups are spinning when all the other boats around me in the marina have stationary cups.
Installation and set up on the Vulcan were pretty straight forward, you just need to find space for the receiver unit, which is the same 'burger bun' size as their ZG100 GPS/compass unit. It's been running for a year now and never missed a beat, so I'm very pleased with it. One thing that I have done is to set the Vulcan to use SOG to calculate true wind speed and direction, rather than the speed from the underwater log impeller, as the log seems to be pretty useless from about 2 weeks after cleaning, and needs about 3-4 knots of boatspeed to get it even to turn. The purists say you shouldn't use SOG for true wind as it can be misleading when there's a strong tide running, but that only applies relatively rarely at sea, whereas an incorrect speed through the water happens most of the time.
Regarding size, it's pretty much the same size as the Garmin, and it doesn't look out of place on a BC26 mast. The support wand is perhaps a little longer than necessary on a small boat, but once it's up there you don't really notice it.
All in all very pleased, and glad to be all B&G. (Just need to get the DST800 fixed now as the echo sounder seems to have learned a trick or two off the Garmin gWind and that disappears for a fag now as well!)
P.S. sorry just noticed that the photo is of the Garmin gWind not the B&G, I'll see if I've got a B&G photo and update shortly.
Update 2 - photo in the harbour has B&G. Sorry I don't have a higher res pic but I think it shows relative size.