I spent about 10 hours driving home eight days after GRS in Canada (that includes breaks, though - it'd be about five hours straight through). I was easily able to sit up in the passenger seat, and I actually drove short legs of the trip, about 30 minutes at a time followed by 2 hours or so of resting. It wasn't fun, but I was fine; I had stopped taking heavy-duty painkillers by the third day post-op, so I was only on Tylenol by then.
The biggest problem was that I had an as-yet-undiagnosed allergic reaction to the Celebrex, so I was unbelievably itchy all over! The actual pain was minimal.
Everyone is different, but I'd say it's reasonable to expect that you'd be off the serious meds and able to lie down without a mattress by 3.5 weeks post-op. (At three weeks and five days post-op, I went back to work full-time, including driving myself there and back. I had some discomfort, and I tired easily, but it was doable.)
If at all possible, I'd recommend that you budget that $500 for overnight stays in motels instead; that way, if you don't need to stop, you haven't spent the money, and if you do, you're going to be "buying" a bed and resting experience that's much nicer than a mattress in the back seat.