On the dirt cheap any stick on chipset cooler is enough to get by with while ones with a fan are nicer, there are still a small trickle of new old stock 486 coolers on eBay.

I am looking for something more effective like a socket 370 cpu cooler. I just don’t know what adhesive to use to fasten it to the cpu. The cpu is not going to be used in any other computer so I don’t care if it is permanently fastened to the cpu.

Well not tape, anyway. It's never thin or conductive enough.

Look into Arctic Alumina adhesive or Masterbond thermal adhesive. Make sure it's specific for heatsinks.

If you are lucky you can also find clamps that aren't broken plastic, or make some if you're handy, and use thermal paste instead of adhesive, which has the benefit of being temporary and removable.

If you find clamps for sale online (like from a late 486 era heatsink), make sure they're the right size as CPU dimensions varied a lot.

A socket 370 HSF would be major overkill in this situation. Even the models of 486 that specifically require a heatsink and fan don't absolutely need anything overly large or elaborate. A proper 486 HSF from that era is honestly no larger than modern-day chipset coolers, as others have already stated. When I was building my childhood 486 recreation a few years ago, I was lucky enough to find a 486 CPU cooler with fan on Amazon, of all places. It doesn't even need adhesive or thermal paste (although I did add a small amount of AS5 just because I had some on hand), it simply clips directly to the CPU itself (not the socket) and works perfectly fine. An alternative would be to take a working heatsink and fan off an old video card and use that (I don't know what adhesive to recommend, as I've never actually used thermal adhesive before).

I generally don't mind overkill when it comes to cooling because there is never any harm in that.