LMR64010 wrong feedback voltage and wrong output voltage
If you have any suggestions regarding what else to check, please do
tell

Check is that your output current is not too high: -

Sure, your input voltage is a little higher at 3.7 volts but, realistically it might mean that you could achieve 115 mA for the output load current at 15 volts.
Look at the losses - with 100 mA and 15 volts (a power of 1.5 watts) at the output, the input power will be about 40% more so, that's 600 mW dissipated in the chip and, that will get warm without significant copper around the device to take heat away.
You might get around 65% efficiency at 3.7 volts of course so do some math and see if your output current is actually demanding too much from this feeble (but nice) little chip.
You should also look at the inductor losses and check that its self resonant frequency is not approaching that of the switching frequency used.
There is also a serious doubt about pin 2 (ground on the chip). Your layout does not show it connected anywhere: -