There are others who could help more than me, but since no one has answered your post, here goes.

There were some very nice sporters built on the Charger Loading Lee Enfield (CLLE) and Short Magazine Lee Enfield (SMLE) actions, but most of the "Lee Speeds" we see were built on Lee Enfield Carbine (LEC) or Lee Enfield MkI/MkI* or Lee Metford barrelled actions. These are the way to go if you don't want the charger bridge.

Since you want to use .303, your best bet IMO would be to find a pre-98 rifle in Canada and have it shipped to you. There SHOULD be no problem getting it across the border if you arm yourself with the BATF letter beforehand.

There are not many such rifles around with bores in a good enough condition for your project though. You could try the major gun parts dealers in the US, as some of them have had new or excellent barrels in the past.

The barrel from an SMLE will thread into the earlier Lee Enfield actions, but the barrel from a No.4 Lee Enfield will not.

You'll notice that most of the Lee Speed sporters have the earlier shape of Nock's Form with a second, higher "step". To me these are more attractive and suitable for such a project. It won't be easy to find one with VG+ rifling however.

Another option would be a new SMLE "H" or heavy barrel from Australia, but this has the Nock's Form for the SMLE for which is was made.

Another option is reboring a worn out barrel. This way you could preserve the original barrel with its matching serial numbers, date and configuration of Nock's Form etc.

I just bought a mismatching 1897 Lee Enfield MkI* made at Sparbrook for less than $50. Matching, but sporterized examples up here can often be had for $75-$200 depending on bore and overall condition.

Many of these butchered "Long Lees" or "Long Toms" as they are sometimes called, have the cut-down remains of their military forends in a good enough condition that they can be easily reworked into a sporter configuration. The quality of the wood was excellent, and the metal to wood fit was also almost of commercial quality, so you save yourself a lot of trouble this way.

If you've studied the photos, you will have noticed that the lower end "Lee Speeds" or commercial sporters used military pattern butts with or without checkering added, so that is another option. If the butt has been sanded or worn down too far to match the original brass butt plate you can either reduce it to match or replace with something else that looks right for that vintage of rifle.

Sights, engraving, bluing etc. you can choose to suite your taste and budget. Have fun!