The filter is not your problem (unless you didn't replace it the first time either), but that you now have the wrong fluid all in your system, and you can't just drain it out, because when you do that only 2/3 of all the fluids will come out. So, you'd still have a lot of the wrong fluid circulating.
What I'd do is drain the fluid, refill it with the correct one, then try to run the car through all the gears very carefully (which now you shouldn't have problem - or at least not major one - doing anymore), then immediately drain it out again, refill... and then do another complete drain-refill cycle in probably a thousand miles or so, this time also replacing the filter. This way you could probably reduce the amount of the wrong fluid to 1/16-1/32 of the total fill, which should be negligible (even if still not optimal).
Obviously if after the first refill with the correct fluid you're still not able to switch gears at all, then either the wrong fluid already did some permanent damage to your transmission, or there's also something else wrong with it - so, there might be a re-think of the plans and getting an actual and real transmission expert involved necessary.
You can get Mobil 3309 for ~$150 in a 20L bottle, which should be mostly enough (or need just 1 or 2 liters plus) to do all 3 drains and refills. I suggest you also replace the drain and fill bolts if you didn't yet, or you risk ruining the threads in the housing. Per manufacturer's recommendation they should be replaced every time, but it's probably an overkill if you're doing another drain(s) in a short time anyway and are only re-using them for a second time.
You'll also need VCDS to do all the fluid changes, because if you're not doing the fill at the correct temperature, your fluid level will be wrong and that will also cause problems even if the right fluid has been used.
