Whether a diver sinks depends on their buoyancy not their weighting, unless they have more weight than their BC can compensate for, in which vase they will be swimming up.
It's an easy guess that a diver is overweighted because the finer points of weighting are usually not covered in an OW class, but whether a diver moves up or down in the water column is dependant on buoyancy control. Overweighting may make it more difficult to maintain buoyancy when shallow, but not impossible.
If it the idea that there is a direct relationship between weight and buoyancy is one reason divers are overweighted to start with. The "if you float you must need more weight" syndrome along with "if you hold are able to maintain depth you are properly weighted". Well I've held a safety stop holding a 15# anchor, not recomending this to anyone, so other than demonstrating risky behavior, does this mean I'm weighted 15# light normally?
In inverting, the OP could be lowering his BC in the water column and making it slightly less buoyant, either by how he inverts and/or by the trim lowering his head and that end of the BC. Tricks like this are not easy and any small factor can cause failure. I agree that the OP should be properly weighted, which may help if he is not already properly weighted.
As far as trim goes, ones weight should be adjusted so one can maintain any position, or just to maintain the position you perfer. Proper trim is more a personal preference rather than a law.
Bob
Just for clarification for others heavy overall is a buoyancy issue, heavy front or back/ side to side is trim. deal with overall heavy or light first and then move weight around to deal with trim. Like real estate trim is all about location location location