TRIAC gate resistor keeps getting burned
The datasheet for the triac says the average gate power is .5 watts. You need a resistor rated for at least that. Also, If we look at the Triac wikipedia it discusses that 3Q triacs are designed to operate heavily inductive loads. This would explain it functioning better. Normal TRIACs or 4Q triacs have problems due to high voltage and current angles when they turn off at zero crossing. They experience a voltage spike which can turn them back on again. Commutations (▲v/▲t) can be improved with the use of snubber networks. They are designed to provide at need to help balance the circuit. With the introduction of 3Q triacs this problem was avoided. If shopping for a device like this you can simply search for "Snubberless Triacs" which is what they are often called.