rm -rf

It sounds like it's might have hardened / polymerized, like varnish, after sitting in the sun. Mineral spirits would take it off when fresh (and works on tar, too). Turpentine would work the same as mineral spirits, so I wouldn't buy any turpentine -- it's mostly for specialized oil paint purposes now. Mineral spirits on a paper towel are great for cleaning up black chain goo, too, so the rest of the can won't be wasted.

Try the mineral spirits first. Testing on the underside of the frame is a good idea, but I don't expect it to affect the bike paint. It's been fine on my bike's paint.

Since the hot water worked a little: apply layers of paper towels, soaked in water. Some dish soap would be good, too. Then wrap with saran wrap, or other food wrap, to keep it from evaporating. Let it sit all day. It won't dissolve the sap, but might get it quite soft.

I've used a similar technique with paint stripper, which dries out quickly. Saran wrap keeps it wet and active for hours.

As a last choice option, paint remover would likely work on the sap. It may or may not affect the original paint. I'd be very careful, testing it on a hidden underside location, very briefly, then a longer test. Look for dulling of the paint, not just paint bubbling off.