Room Air Conditioners…

I have a plug in electrical tester that indicates correct wiring, when plugged into most outlets the left two yellow lights are lit, however I have this one round outlet for a 20 amp circuit for an airconditioner, when I plug it in all three lights are lit

I have a 8,000 BTU portable air conditioner, and when the compressor turns on all the lights dim.
I looked in the circuit breaker and its on a 15a fuse, which for some reason is wired into the kitchen lights/fans, as well as one of the bedrooms lights/outlets and half the outlets in a second bedroom.
lots of computers, tv’s, and other equipment are plugged into these outlets so it would be difficult to get the air conditioner on its own circuit.
Is there anything else I can do?