Not sure either of these helps with your budget question, but I have used them both with the JLA F110 to great results, and I offer for your consideration. Pass Labs makes a great external crossover, the XVR1, which although a bit complicated to set up, once you get it right, its terrific. Look around for a used one (or try Reno). Also the McIntosh MEN 220 room correction unit is quite good, and does a lot more than just cross-over to your sub. Again, find one used and its a lot of value add for the $.
Finally (and separately) and with all due respect, as to the suggestion above from a member that you should drop the sub because your "SFs completely out class it;" well, that is complete nonsense. I'm not comparing SF with JLA in saying this, what I'm saying is that having a great sub (which you have) well integrated into your system will, in fact, enhance the overall performance and experience. Your SFs are very nice to be sure, but they will be even nicer with the sub, assuming it is properly worked into the mix.
Finally (and separately) and with all due respect, as to the suggestion above from a member that you should drop the sub because your "SFs completely out class it;" well, that is complete nonsense. I'm not comparing SF with JLA in saying this, what I'm saying is that having a great sub (which you have) well integrated into your system will, in fact, enhance the overall performance and experience. Your SFs are very nice to be sure, but they will be even nicer with the sub, assuming it is properly worked into the mix.