I understand the allure of an original Old Model 3 screw Medium size 357 Blackhawk in 44 Spl; either the flat top frame or the ribbed rear sight top strap.
What you propose is very common and has been done many, many times with excellent results. Re-boring the barrel is about $130 + shipping. Reaming the cyl to 44 is under $100. Using your original barrel and cyl saves you money because the barrel can be re-installed w/o needing machining because the bar/cyl gap will be unchanged when re-assembled.
If you can't do it, someone to pull barrel and re-install is another $100 or so + shipping. These are just ball park prices to give you an idea of cost.
Below is my OM FT completed conversion to 44 Spl, all steel including grip frame.
OM Super Blackhawk 44 Mag barrel, cut to 4 5/8" with resoldered front sight.
Ruger New Model 44 Spl factory cylinder, only now available because of the NM FT mid-size Blackhawk 44 Spls. Found on eBay.
XR3RED Old Army stainless steel grip frame with SBH wide trigger.
Another Old Model mid size frame with ribbed rear sight top strap, SBH hammer, SBH wide trigger, and stainless new model XR3-RED size stainless grip frame: