at that point it will chug way down to about 500rpm for about 2 seconds and then it will rise back up to the high idle.
This (still) sounds like to me the fuel bowl is empty (not the accelerator pump circuit) and it takes a second or two for fuel pump to refill the bowl.
The accelerator pump circuit uses check balls and weights to keep that system air tight (so to speak) so you generally will have 2 to 4 pumps before that circuit go dry – if bowl is dry.
A couple of ways to test (unfortunately there is no easy way).
1 - Keep cycling the throttle and watching the pump spray. If you got 4 shots then no additional good sprays (then bowl is empty). If fuel bowl is full, then you will have roughly 50+ shots until empty.
Be sure to leave throttle valve open a bit, because 4 or more shots can/will flood the engine.
2- remove carburetor and turn upside down over a container that you can use to measure how much fuel is inside it on a ready to use cold start engine.
3 - Remove the carburetor mounting bolts and choke clip. Insert cardboard under the cold start carburetor and check how many full squirts you can get (doesn’t flood the engine that way - but still messy).
4 - remove carburetor top (on engine or off) and see if enough fuel is inside to keep float up against the needle and seat. Sense float is attached to top – and no longer in the bowl, I would guess the remaining fuel would be ½ to 2/5th full. If much less than that, then you have a problem.
BudW