There is a saw that will cut in tight places, but it is not a woodworking tool, it's a medical tool for cutting bones and the like. As a past orthopedic PA, I became aware of its use for some wood-working jobs in very tight places. These operate by compressed air. Here is a link to one type of saw:
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a search for orthopedic bone cutting saws and you will find other companies that make these saws. I thought Midas Rex made these, also, but I didn't see any on their site. Ebay may have some used ones, but these are outrageously expensive.
Also, a cast cutting saw will work the same way and may be much cheaper. A cast cutting saw is also a reciprocating type saw. It usually has a circular teflon coated blade, but bone cutting type blades can be installed on it, allowing for getting into tight spaces for cutting. I have a used one (we had in our medical office) that we replaced because a bushing wore out (which I fixed). I have reason to use it, maybe, once a year. Twenty years ago a Stryker cast saw with vacuum attachment went for about $200.00. I see today (I just did a search) just the saw, new, is about $900.00. WOW, that's some difference! Maybe a used one can be found on Ebay for a reasonable price. I should go into business reparing broken ones. They are easy to fix, few moving parts, etc., but alas I'm retired, so.....
I dare say, I doubt if my post is encouraging for you, with these prices noted. Maybe there is a similar tool, for woodworking (that I am not familiar with), that is more cost effective.
Sonny