This is back-seat-driver, off the cuff stuff, with-no-experience-in-these-things, but:
The vendor that produced this card likely a) has their FPGA instructions stored in a memory chip, and usually these are re-programmable (aka 'firmware update'). If not, it means they're b) loading the instruction set on the fly, usually as bytecode contained in their driver software.
If the card happens to have an open-source driver, and you're really lucky, it may be possible to figure out either a) how to update the firmware with your own instruction set, or b) figure out how they load it on the fly, and load your instructions instead.
If you're unlucky, they don't have any open-source driver or equivalent, or the bytecode they use is signed in a particular way to avoid people doing exactly what you're trying to do. This is more common in, say, a smartphone than an off-the-shelf video recorder card. In either of those cases, your only choice is to interface with the FPGA directly as EM suggests.
Regardless, it's going to take some work and a lot of trial-and-error to get anywhere with it. By the time you do, the 10-15mh/s may no longer be worth the power it consumes.