Looks like I wrote a novel so TLDR:
I suck at this stuff, just getting back into custom ROMs - tried Pixel Experience 13 and Lineage OS(no Play Services (I think Android 13/A13 Beta)).
Stock Sony Rom + Root is the way to go. At least for right now.
Wouldn't count on GCam working or working well.
(Sony) Photo Pro is proprietary too afaik so you'd need someone to port it to custom OS like Lineage without G Services (what I read someone feel free to correct me). It might work with LineageOS with G Services or whatever else, I didn't try it.
END OF TLDR:
Pixel Experience was weird, probably worked fine but I don't think the dimensions of the Xperia 5 III work well with how PE is scaled. Just this users opinion. Also I really only tried it to try it- not a huge fan of Google lately.
Lineage OS almost worked but the version I had text messages didn't work. The default one or others from F-Droid. I'm positive its user error on my part. Battery life seemed great though, it's anecdotal but it seemed markedly better without Google constantly checking in. Probably my bias.
Its important to me to have some Sony Apps and overall functionality (seems like low dev support when comparing to other devices IE Galaxy/Pixel) so I think stock Sony ROM+ Rooted is the way to go. Seemingly the general consensus on the forum too (especially since its pretty stock Android anyway).
Bonus objectives for me on stock Sony Rom is to DeGoogle + MicroG although I haven't made it all the way there yet. Then maybe long term try a Xperia 5 IV ROM for sh*ts and giggles for Android 13 since I doubt the 5 III will get it.
Your questions about GCam and losing functionality: I have heard/seen a lot of things online about GCam either needing to be ported to devices individually or that its part of Google's "Special Sauce" software wise. So I would expect stripped functionality if functional at all.
Someone who is a lot smarter and experienced than me should clarify on these last two:
Downsides to an unlocked bootloader+Custom ROM is same as always: risk of user error, bricking your device (soft bricked mine), depending on the ROM you choose you may not have Google Apps/limited so Banking apps and streaming apps won't work or won't go all the way up the resolution chain 4k20 69 fps etc. And I'm sure there is a security disclaimer somewhere out there too if you aren't careful.
Pros are same as always: Control of your device, system wide ad blocker, admin rights to delete bloatware, customization, custom ROMs, better battery life, potentially longer OS support, privacy, security, etc.
RAMBLINGS OF GETTING BACK INTO ROOTING WITH THE X5 III
*In case there are others out there who are similar to me in either getting back into rooting after a long time or the first time*
I'm pretty much a novice at this stuff but I was finally able to get my phone unlocked a few weeks ago and tried a few different ROMs. Out of what I tried I really only remember the Pixel Experience and Lineage OS. PE was fine but I'm not a huge fan of all the Google stuff so I went in the opposite direction and did Lineage OS without the Google Play Store/Services/App Framework whatever. It was almost a keeper for me except messaging wasn't working for whatever reason ((I wonder ^^)but really I think it was because it was an A13 beta). I tried a few different messaging apps but between that and losing some Sony Apps I decided to back to stock.
I feel like the big points for this phone are the camera app, the display customization depending on the content viewed, and possibly the gaming tweaks/music app? Last two are a stretch but eh ¯\_(ツ)_/¯.
The new plan was/is hoping to do was DeGoogle the stock ROM and replace it with MicroG. I haven't gotten around to it because the post I was looking at on how to do it on this forum linked to another post in a different forum and it was for Android 11. My phone is on Android 12. Looking at the comments people had said the process was the same for 12 as 11 or very similar. Shouldn't be too hard but knowing me it will take me another afternoon or two to get it figured out and working.
Long term I'm kind of hoping that when an Android 13 ROM comes out for the Xperia 5 IV we could try throwing it on our devices too. Since I doubt we will be getting Android 13 on the Xperia 5 III. I know the physical camera is different on the 5 IV but everything else seems close enough to a normie like me that maybe it would be worth it to try. Then its back to figuring out how to DeGoogle on an unsupported device and installing MicroG etc etc.
Sorry its such a late response and not the most technical (or maybe even 100% accurate) but that was at least my experience a bit ago trying a few different ROMs on the GSI thread linked in the "How To" post listed here.
This phone has really grown on me and I'm bummed I'm not savvy in any amount of Android development so I am unable to make customizations/tweaks to the stock ROM. Or customize other OS ROM's to port here. Which makes me reliant on others to do the work so I can leech off it haha. On more widely used phones like Samsung or the Pixel its NBD since there are so many users. But a small community like this, it doesn't seem like there is a lot being done. Not a dig or anything, just not a ton of options. Although being able to install a ton of different custom ROMS from the GSI thread is really cool even if it isn't tailored to the Xperia 5 III.
Part of why I threw up so much info on here goes with the last paragraph. You probably won't get a lot of responses. There aren't a ton of peeps here and those who are generally already know what they are doing.
If you end up trying a bunch of different ROMs and find one you really like LMK!