To answer OP's original question, I would probably advise against it. That much inpatient time will almost undoubtedly affect your research (even 1 day of outpatient clinic per week can be a lot if you're going down a physician-scientist route). Not to mention that once you start the hospitalist gig, you're probably no longer seeing outpatients which is where the bulk of your clinical practice will be.
You probably would need a K or something similar (ASCO CDA, Damon Runyon) in most instances unless your mentor(s) are willing to pony up some of their funds to cover your salary, or if your institution is ridiculously supportive. There really isn't a good answer IMO besides trying to find a well-funded and supportive mentor (unfortunately Venn diagram overlap for those isn't 100%). If you're unable to get any funding, my general sense is that you're usually SOL. Options are to lean more heavily into clinical research, or continue collaborating with a lab PI as their "translational" person in clinic. I've known people who've done those and are very happy with their current roles. Alternatively, I know a lot of people who are currently in this position and have gone into industry or are considering it. I'm in academics myself, so I can't speak to the industry path, but I do think we get brainwashed a bit into thinking academics is best when there is a whole world of other career opportunities for people with both clinical and science expertise. The people I know of who went into industry are also (currently) very happy, though to be fair they've only been doing it for a few years