The spits hadn't been "taking it to the LW" so much as "holding their own." The LW was more than a match for the RAF after the BOB. Losses were sustained on both sides, but looking at JG2's performance over the channel, they were downing Spit5s by the dozens every month. 109F2, F4, G2, and 190A2/A3/A4 in 1942/43 were superior to most of the RAF planes in overall combat effectiveness. The RAF was not responsible for breaking the back of the LW IMO. They repulsed them. They stopped their plans during BOB. They didn't break it. It was still more than strong enough to fight on a 2 front war for years afterwards.
Staying over the UK wasn't going to stop the war, and wasn't going to break the LW. The spits lacked the real range to do much other than nuisance raids and circuses over France. The early P-47s over Europe saw widespread use beginning in late 1942 and early 1943, and kept pounding on the LW until it was broken by the time P-51s showed up. The P-47 has more kills than the P-51 in europe, by several times last I recall. They were the ones pushing into the continent because P-38s were shunned (won't get into that) and P-51s weren't ready, and the RAF compliment couldn't make the range.
Later on when the LW was being pushed back and the ponies were showing up to escort, the P-47s would often go hunting for the LW and even attack them on the ground, further depleting their numbers.
Like the hurricane in the BOB, the P-47 over Europe did most of the work and the P-51 gets all the adoration. They both did well, but one had a larger hand in the overall success.