Can I slow cook in short intervals?
The obvious downside is safety: each time you go through a heating and cooling cycle, your food will spend time in the danger zone. It's hard to guess exactly how long, but if it's a large volume and it takes an hour to cool to 40F and 15 minutes to heat back up to 140F, and you use the conservative end of the safe time range (2 hours) you might have a problem. You could partially alleviate this by using an ice bath to chill faster.

If you can avoid the safety issue, in terms of the resulting broth, it sounds fine though.

That said, slow cookers are awesome, they're not too expensive, and that'd solve your problem with much less effort.