Yes you can get more analog inputs by connecting an AD or AD/DA converter via ADAT.
ADAT supports 8 channel @44.1/48kHz and 4 channel @88.2/96 kHz and 2 channel @176.4/192 Khz by channel multiplexing.
But are you aware of, that the BBF Pro FS supports already up to 4 analog inputs ?
If you want a very good RME product, then you could take e.g. the ADI-2 FS:
https://www.thomann.de/de/rme_adi_2_fs.htm €685
8 additional channel would be delivered by the ADI-8 DS MKIII
https://www.thomann.de/de/rme_adi8_ds.htm €1466
Alternatively, if you want at the same time an excellend AD/DA converter and phones preamps / USB DAC with advanced features: https://www.thomann.de/de/rme_adi_2_pro … dition.htm €1695
I know it should not be so expensive as you said, I only wanted to give you this overview whats possible.
So the ADI-2 FS sounds like a feasible solution to you.
Additionally it gives you two very nice phones outputs and has a nice compact form factor.
BR Ramses - UFX III, 12Mic, XTC, ADI-2 Pro FS R BE, RayDAT, X10SRi-F, E5-1680v4, Win10Pro22H2, Cub13