Your "bright drawn mild steel" would probably be a low carbon steel that was drawn thru a die to give it a good finish.
If so it would be similar to our 1018 or 1020 which are low carbon steels.
Speaking of cold finished wrought low carbon steel (not DOM tubing) they usually have a tensile strength of around 62,000 psi with a yield strength of around 53,000 psi.
Either of these will work all right for the pressures found in muzzleloading black powder guns.
If the mechanical properties of your "bdms" are similar then it should be fine.
As for proofing your barrels, as you live in England you may have to have them proofed at a Proof House? If so, their set up to do the job and conduct the necessary tests following proofing.
An individual, in my opinion should not proof their own barrels unless they have access to someone who can conduct Magnetic Particle Inspection or Fluorescent Penetrant Inspection (at the very least).
A post proof inspection is needed to assure that a internal flaw was not created by the test. Just looking at the barrel by eye is not sufficient.