Don't know where you I've so going to give you the info on schools we considered in Sussex/Kent.
Lancing does have Saturday school & an expectation that all pupils will attend Sunday church services every few weeks. It was DS's original first choice. He changed after viewing another school. Around 60% board but most are local & flexi boarding could be an option as they allow day pupils to stay over regularly. DS has many friends who attendLancing or who have recently left. Very much a traditional public school with solid academics. We found it a bit too stuffy for our family & were secretly pleased when DS changed his mind on this school.
Hurst I believe is Monday-Friday. Excellent sports, Not as academic as Lancing which is not as academic as Brighton. That said, you get bright kids at all schools.
Didn't like Lewes Old Grammar. Very poor communication to start with which completely put us off & we didn't even view the place. Plus when we asked about Dyslexia support, we never heard back from them. Their loss.
Battle Abbey is another Monday-Friday. DS has friends who went there & who moved to other boarding schools for sixth form.
Tonbridge - all boys & don't think they have Saturday school? Presume they have pre sits as well which you may have already missed,
Sevenoaks are a six day a week school & their attitude is too businesslike for me. I didn't want my son to be a client, he's a pupil/child!
Eastbourne has Saturday school or did when we looked at it a few years ago.
Ditto Ardingly
Bedes is always going to be the school I rave about. About 40% boarding now (the houses are fabulous) & they have weekly boarding. flexi boarding would only be available if they've space I guess? Non selective, but fab for bright pupils. DS left with a clutch of brilliant GCSEs. He was awarded a very generous bursary & they were happy to increase if DS wanted to stay for sixth form. It was Saturday school that put him off which they've now abandoned. Excellent bus service for Kent & Sussex which DS Sometimes used to get home on Saturday. Good for sports, the arts etc.
Brighton very academic & a Monday-Friday mostly day school. Didn't like the HT or the ultra pushy attitude. DS pressures himself enough already.
Good luck with your quest OP. When it comes to bursaries, if you want your child to attend, do what it takes. I practically begged & I'm not ashamed to admit it as it worked. My DCs Would never been able to attend private schools without their bursaries.