If you're a ham, you need to learn how to figure this stuff out.
I can't tell from your post whether you measured stuff, or just guessed. If you can't measure volts and amps, your first priority is to buy or borrow a meter that will. So don't get offended If I go too slow. Divide and conquer is the strategy.
First, check for operator error. Is there a current limit control somewhere in/on the power supply that's mis-adjusted? It's possible to have wire that's too small and a couple of inline fuses contributing to the voltage drop. Measure the voltage under load at the power supply terminals and as close to the handheld terminals as you an get. Also consider the possibility that if your handheld battery is dead, the charging current plus the transmit current may exceed 4A. I'd say, not likely, but worth a think.
Next thing I'd do is check the power supply. Use a big resistor or a car headlight or a couple of tail-lights to put a 4A load on it. Measure the current and voltage. If you get rated current at rated voltage, it's probably good. If not, fix it. If you just bought it, take it back.
Does the VX5 have normal battery life when run on batteries? That shows that the final is probably not blown drawing excessive current.
But it is possible for the final transistor breakdown voltages to have been reduced by overheating and draw too much current on 12V. Verify the external power supply load by running it off a car battery and verify with your ammeter that it doesn't draw more than 1.9A. If it's drawing more than 4A, it will drag down your power supply and probably go up in smoke. Maybe a 2A fuse during the test to reduce the smoke.
Ok, so they're both good...must be interaction. Put the 4A lights on the power supply. Run the HT on internal batteries and waive the rubber ducky around the leads to the lamps while transmitting. If the lamps dim, you know you have RF getting into the power supply. Try output bypass caps and ferrite cores on the wires. If that don't fix it, you'll have to get out the radio amateur handbook and learn about keeping RF out of your power supply. This should not be a problem with any modern commercial power supply.
It's almost never a good idea to parallel power supplies. It's always a BAD idea when one of them is broken. If your 4A supply ain't broke, you don't need to parallel. If it is broke, you need to fix it.
If your parallel lashup gets more broke and puts out 20V, you'll have a handheld pile of burnt junk.
mike