Back from a deep sleep. Dug out and settled my Mega PC again after moving out and thus having much more room. I'm glad to inform that I found a valid solution for displaying both the PC and Mega Drive parts of the Mega PC on any kind of monitor. I tried many solutions, including a Gonbes 8200 (slightly blurry and imperfect line-doubling, even at 60Hz), an Extron DVS 304 (only games running at 256x224 and 256x192 would show a picture...) and a Wei-Ya ACV-011.
It's the last one who gave me the best results. It can be ordered here ($50 with free shipping) among others: https://sintron-hk.com/Wei-ya-ACV-011-CGA-to-VGA-video-converter-P2699910.aspx
What this unit does is clean line-doubling: 15KHz transformed into 31KHz. Results isn't perfect yet (picture distorted at the top and some interferences are to note sometimes) but picture looks really nice IMO. DIP switches settings that give the best results are setting 14, OOXO. Setting 16 is to avoid as the top of the picture gets funnelled and the picture randomly disappears for 1 or 2 seconds.
Some pictures here: https://imgur.com/a/SrJhZ
Also, I took the opportunity to upgrade my unit with an 486SLC motherboard too! Except... there's a little problem as this is an earlier revision with an identical layout to the original 386 one (same chips everywhere, including the Western Digital 90C11, but with the 512K of video RAM already built-in) and only running at 25MHz instead of 33. There is a performance increase but it's not as drastic. That "new" motherboard has some incompatibilities if using an SB16 Value sound card. You can't use low DMA at 1 or some games will play their sound effects with a stuck loop (I had to set it to 3). Another minor issue, the PLAY.EXE audio player (from the SB16 software itself) randomly crashes the machine when attempting to play a 16-bit WAV file after an 8-bit one now.
The other good news is that my original battery didn't leak despite letting the machine off for way too long. The one from the 486SLC motherboard didn't leak either but was dead. I did my battery replacement job (with an Amigakit CR2032 holder) on the 486SLC one and left the original motherboard with no battery.
For the info, the 486SLC motherboard comes from an Amstrad PC7486SLC80 base unit (where the 80 is the HDD capacity, not the clock speed!) I bought from eBey.
I might add some other pictures to that album later. I also hope to upload a video of the whole disassembly, motherboard and battery replacement process but with my usual laziness, it's not that easy.

Edit: Just thinking about it now, does anyone here happens to still have all the software pre-installed on the Mega PC's original hard drive? I'd like to put my hands on that cool front-end menu (there's a screenshot of it on the PC7386SX manual cover) and that tutorial that I used to play around with younger. The only thing I could find till now to download is Counterpoint (I still have the SVGA utility disks, as far as I remember).