My G1W-H started having trouble holding the correct date and time for more than a day, so I pulled the capacitor from my old broken G1W-C and placed it in the H. I just replaced the battery in my other H, and am too impatient and cheap to order another battery and wait for it to ship. It took a bit of trial and error to successfully update it to a G1W-C firmware. For any firmware update on an H, you have to change the file name to either FWDM800H or FWS1088. Neither the 2014 or 2015 version worked properly, as predicted by previous responders. The screen stayed completely black with both, but the camera did apparently record video with the 2015 version. I ended up installing this firmware from 2016, which works so far.
https://wr24support.freshdesk.com/helpdesk/attachments/8003269087 The menu and record buttons were flipped, however, which can get a bit confusing. The real problem is that it does not appear to shut down properly. The screen just goes black immediately without that characteristic sound. This leads to the dreaded last file corruption, which defeats the purpose of updating the firmware in the first place. If anyone knows of another firmware version that I can try, please let me know. Is there a firmware for the HC out there perhaps? If all else fails, I'll probably end up going back to the H firmware so it can at least have the correct button mapping, and maybe looking for a replacement battery in stores.
Edit: It took me multiple tries plugging and unplugging to get the camera to turn back on. It turns on every time now, but I've lost the date again and all settings have reset to default. I'm starting to wonder if the battery was even the problem, or if this firmware is just letting me down.
Edit: Every single time I powered it back on, the settings were back to default. The date remained intact though. I ended up giving up and flashing an H firmware, which made no difference. Now it's losing the settings AND the date every time. I may just end up calling it and putting the original battery back in while I look for a suitable replacement battery.