Hockey,
I would try a Bulldog with the load I want to carry first.
I have a Bulldog and with some of the more powerful loads, it can be a painful experience. I know that everybody likes to say what a great, easy recoiling and powerful round the .44 Special, but they usually are referencing that with the S&W model 24 or a full size single action like the RUGER Blackhawk. In my model 24 and my .44 magnum, those cowboy loads that are still available, are like shooting .38 Special's in a S&W model 10 or 15. Pleasant and they make a bigger hole on "PAPER". The old "KEITH" style loads that made the .44 Special's reputation as a man or bear stopper might damage your Bulldog or your hand, maybe both. I have shot the 246 grain lead round nose (lrn) in my Bulldog and the recoil is not bad, but the round is weak and less effective, if not much less effective for self defense than a 9m.m. with a premium bullet or even my .38 Special loaded with the FEDERAL HST load. In fact, I would prefer just about any +P .38 Special hollow point for self defense over the 246 grain rnl .44 Special.
If you upgrade to more powerful and effective loads, your recoil goes up. I used to shoot WINCHESTER Silver Tips in my Bulldog, but think the CCI 200 grain jhp is a better choice. Still recoil is noticeable and this is a mild load. I tried a .44 Special 250 semi wadcutter load going about 900 fps once. Never fired that load again in the Bulldog.
When the Bulldog was introduced, the standard pressure .38 Special rnl bullet was still the round carried by most people carrying a snub nose .38. It was WEAK and the .44 was a general improvement. With the introduction of more effective +P loads in the .38 Special, you can do more or at least as much damage using a +P .38 Special without the heavy recoil.
I love shooting .44 Special ammo in my S&W model 24 and it should be fine in say a RUGER GP-100 or S&W 69, but I think the Bulldog is too light to effectively use ammo that is more powerful than the best .38 Special +P self defense ammo.
You may not be as recoil intolerant as I am, so you want to carry that ammo, ok, but try them out side by side with a compact 9m.m. like the GLOCK 26 or SPRINGFIELD ARMORY XD9 Sub Compact (my choice) and see which works better. I know which one works better for me.
JIm