The best thing to do is get a new cooler. I also understand the costs and why that not practical.
I would say to rinse it as best and you can with something that cuts oil well like kerosene. It won't effect the oil. Then blow out and rinse with engine oil well.
Now the most important part. What side of the cooler was the filter on? It should be before it. If so great. If not, not bad either. If it's was before the cooler with engine oil, and still will be there then there is probably nothing in the cooler. But just to be safe I would temporarily fit a secound inline filter in after cooler as a last chance catch before it goes back in the trans.
Inportand note. I don't now what type of trans you have so I would not run the car vary hard with the this secound inline filter. It is going to ask a restriction to the amount of oil that goes back in the trans and could effect a lot of stuff in there in bad ways. Only do this for the first 1/2 hour or so to catch any thing that may want to break loose. Then go back to the filter that the trans manufacturer recommends.