Driving a low current solenoid with a low voltage H-Bridge
So let's talk this out. An integrated H-bridge circuit is going to do one of two things - it is going to apply it's own supply voltage to the load (for instance, the DRV8836), or it will give you four electronic switches with integrated drivers to connect however you like.

In either case, your current source becomes a problem. The totally integrated module isn't going to work without a constant supply voltage, period. The other module still needs a constant control voltage, but the source could be attached to the switches as you please. The problem is, without really careful design work, the constant current source is going to attempt to keep pushing through the switches when they transition from on to off, potentially destroying the module - so, a change in direction of current flow now requires turning off the source, changing the H-bridge configuration, and then restarting the source. Perfectly workable, but a cursory search hasn't yielded a good module for this yet.

If it were my design, I would compare the cost of any module I found that was suitable against the cost of simply building an additional current source with the opposing direction. A finished, contained project was going to have to have a controller of some type anyway, so I would then abstract away timing and control in firmware. At these currents and voltages, I could imagine accomplishing this with a 5 volt power supply, an arbitrary microcontroller, a couple of diodes, and possibly as few as one quad opamp chip, depending on how clever this really needs to be.