Your using a part for a non-standard application, so you have to buy and try. In reality, you need a few power supplies and the ability to set the logic levels and the ability to check whether 1 mA is good enough. Application engineers can also help. Selecting a low current output driver increases success. Selecting a MOSFET bridge increases success even further. Parts availability is yet another.
Except for the bridge, the design is relatively straightforward.
To support the low current (1 mA) a MOSFET bridge would be more suitable. Why? Just because the MOSFET is a voltage controlled device rather than a current controlled device.
Now to throw a big wrench in the fan with a totally off the wall idea is to use an optocoupler as the output device.
https://www.electro-tech-online.com/custompdfs/2011/06/4N26.pdf
or a FET version
https://www.electro-tech-online.com/custompdfs/2011/06/H12FH11F1M.pdf
Higher current FET relays are available as well.
This kills the the compensation for the error terms that are giving the most headaches because the transistor or MOSFET is controlled with light. Drive is now cake.
If I were you, I'd build this in two pieces. the driver and the timer.
I hope you tried manually, the constant current source.
Being totally wierd, you could use two power supplies and two relays (optomos or conventional) and 3 electrodes. (3 electrodes, so deadtime isn't an issue)
Optomos relays:
http://www.clare.com/home/pdfs.nsf/www/CPC1788.pdf/$file/CPC1788.pdf
I like to put all the ideas on the table and then kill them one by one.
As I see it, you want to build one and you want to build it quickly and reliability isn't that important and neither is size. Dead time or the amount of time before polarity switches may be a problem. e.g. ON(fwd), OFF(100 uS), ON(REV) may be even more reliable. With 3 electrodes, it doesn't matter. With such low currents, it probably doesn't matter.
Leakage currents, from the devices that are OFF havn't been addressed.
My vote:
Four FET Optocouplers or OPTOMOS relays. The latter is more expensive.