Low cost and Trap rarely show up in the same sentence, Terrance(G)...
Unless you and your wife are the same size and have the same shooting style, you need two shotguns,probably. I'd get hers first, speaking from a vantage point of 24 years of marriage.
I suggest trying out various shotguns, and going with what she likes for that first gun.
I'm using a trap model 870, but plenty of field guns get used for this. The requirements are...
Durability. A trap gun gets more use in a month than some hunting shotguns get in a year. Longevity is paramount.
Weight. A trap gun needs a bit of weight to smooth the swing and cut the felt recoil.Mine runs about 8 lbs, and many run heavier.
For the budget, a pump gun like the 870 is darn near the only choice. Lots of records have been held with the 870, and more will be made.
You might squeeze a field grade 1100 Remington into that budget, but that's a bit doubtful.
Accessories needed will include eye and ear protection, a good recoil pad, and a vest is a good idea. Learning to reload your empties is also good. You probably won't save any money, just get to shoot a bit more.
HTH...
Fit, good trapshooters get really finicky about the gun fitting properly. Ideally, you want one that is stocked and set up to put most of the pattern above the point of aim, tho flat shooting guns are used by some. If it hits high, you can "float" the bird above the bbl and see it better.
Choke. On the tight side is a good place to be, Modified will work fine for 16 yard stuff, and as you move back for handicap, then either full or extra full does the trick. Obviously, a tubed gun is more versatile here.I'm using a Full choke, however, on the premise that it'll make me hold tighter. I'm swapping a few unbroken birds now for a steeper learning curve, but this is a personal choice.