You're right, weight shouldn't matter, and it doesn't at the planes maximum level speed, where thrust and drag are equal. Having a higher top speed helps! Below that speed, thrust helps the acceleration and above that speed, there is a net drag force. What those forces are in proportion to the weight of the plane will determine how much it helps or hinders dive performance. Some examples...
I am going to equate engine power with thrust and drag for this example. They are not equal, but are related, and should be adequate for 'ranking' diving ability and not figuring exact dive accelerations. A plane that needs more power to hit a certain speed has more drag.
P-51 - 365 mph, 1700 hp, 9600 lbs
SpitfireIX - 336 mph, 1700 hp, 7300 lbs
109G10 - 340 mph, 1800 hp, 7300 lbs
Fw-190A8 - 355 mph, 1700 hp, 9100 lbs
F4u-1 - 360 mph, 2250 hp, 11700 lbs
First, adjust the hp(drag) figures for set speeds of 200, 300 and 400 mph (V/Vmax)^2 * hp). The acceleration is the (surplus power + the weight) divided by the weight.
200 mph numbers:
P-51D - 510 hp, 9600 lbs, a = 1.12
Spit IX - 600 hp, 7300 lbs, a = 1.15
109G10 - 620 hp, 7300 lbs, a = 1.16
Fw-190A8 - 540 hp, 9100 lbs, a = 1.13
F4u-1 - 690 hp, 11700 lbs, a = 1.13
The 109 and Spit have the best dive acceleration at this speed.
300 mph numbers:
P-51D - 1150 hp, 9600 lbs, a = 1.06
Spit IX - 1355 hp, 7300 lbs, a = 1.05
109G10 - 1400 hp, 7300 lbs, a = 1.05
109a8 - 1215 hp, 9100 lbs, a = 1.05
F4u-1 - 1560 hp, 11700 lbs, a = 1.06
Pretty even field, F4u, 190 and P-51 starting to catch up.
400 mph numbers:
P-51D - 2040 hp, 9600 lbs, a = 0.96
Spit IX - 2410 hp, 7300 lbs, a = 0.90
109G10 - 2490 hp, 7300 lbs, a = 0.91
109a8 - 2160 hp, 9100 lbs, a = 0.95
F4u-1 - 2780 hp, 11700 lbs, a = 0.95
F4u, 190 and P-51 are pretty close and increasingly superior to the Spit and 109.