Cost vs quality and performance of H-bridge
I would look at small/cheap NMOS/PMOS FETs, for example: AO3400 / AO3401

These can handle a very large Id (more than 4 A) which is overkill for your application but means that the current that the Body diodes can handle is also increased (compared to low current MOSFETs). The body diodes in the AO340x can handle 2 A so in my view they can take over the task of the back EMF diodes easily.

I would ideally need gate resistors AND pull-downs.

That depends entirely on how your H-bridge controls the MOSFETs. If you're controlling the MOSFETs directly from a microController then you only need a high value resistor between gate and source of the MOSFETs. That is needed to keep the MOSFETs off when the microController initializes and its outputs are high impedance / undefined.