There's nothing wrong with using copper wire as a heating element, but I would be concerned with the cats possibly chewing it into a non-functional state.
A long time ago I had this strap-on electric car seat heater. It plugged into the cig lighter and worked like a charm. I think it consumed 4A (50W) in total.
Eventually it failed, and when I ripped it up I found closely spaced slings of cheap thin PVC insulated copper wire neatly stiched to the fabric.
A 30 AWG has a current carrying area of 0.051mm2 and a resistance of 0.34Ω/m.
A current of 0.5A could be considered safe under all conditions (unless it's rolled into a ball, where 0.2A would be safe), and 3A could lead to a wire temperature of 60 C.
So let's say you want to run a relatively safe current of 2.5A (at 5V). That's 12.5W and a resistance of 2Ω, and thus a length of 6m is needed.
So in my view your estimate of 10' in each bed is spot on, if you connect them in series. That gives only 6.25W per bed, which is almost nothing.
You probably meant to connect them in parallel, producing 50W per bed. I suspect this could be too much, and it would require a very expensive (& hot) pot.
The BTU must be one of the most cumbersome modern units made btw.. 200 Btu = 58.6 Watt-hours, not a very good measure of power..
Now, the big question is how large are those beds, and of course if the cats will be comfortable with the power density (W/m2) thus produced.
If you were to halve the current (1/4 of the power) you'd need to add a 2Ω 3W resistor. A 12.5W 5Ω pot for manual adjustment would be more than adequate.