The Metropolitan Police was chambered in .38 Special. The Lawman was chambered in .357 Magnum.
Other than the caliber, the two models are precisely the same.
What will chamber in the cylinder should determine which model you have.
I can assure you that Colt didn't install an adjustable sight Trooper Mark III barrel on a fixed sight frame.
This is almost certainly something done by some Billy Bob.
Your first move should to be to look for any signs of damage to the frame, like scarring, filing on the front of the frame, or a bent or cracked frame.
It's not unusual for a revolver re-barreled without the correct frame wrench to be cracked through the frame right under the barrel threads.
If the frame is not damaged, you could have Colt install the correct barrel and restore the gun. Their factory records will indicate what model you have, Metro or Lawman.
You may have to buy a barrel, since Colt probably no longer has any.
Installing a barrel properly requires considerable tooling, so this is not something you should attempt yourself.
The old gag of locking the barrel in a vise and using a hammer handle or 2x2 through the frame is how frames get bent or broken.