The .303 Lee Enfield has been used successfully to take every type of game on the planet, including elephants. Now, having gotten that out of the way, the SMLE was a great battle rifle and, for its era, was the base for some nice sporters. But, as noted above, this rifle was designed and built in the earliest days of smokeless. While the No.4 Mk I was a WWII redesign, the key features of the redesign were about ease of manufacture and diminished cost. They were not significantly strengthened and the .303 round is not, by modern standards, very hot. If the barrel were worn, a 30-40 Krag would be a good rebarrel option, but the cost and issues involved in going up in caliber are significant for very little return.
If you want to build cheaply, look at the H&R Handi rifle platform. With SB2 receiver, you have actions and barrels that are fine up to 300 Win Mag level (though your shooulder may not be). Because of their design, you can rechamber (ream) quite easily, often with a rented hand reamer at home. You can barrel stub to virtually any non-belted, non-magnum rifle round. 357 Maxi, 356 Win, 45-90/110, 50 Alaskan are all popular rechambers. Barrels are available in a broad array of calibers from .17 up through 500 S&W at under $150 factory fitted, and less if you can do a little home smithing. An Sb2 receiver with generally sell for @ $100. There are a wide array of finished stocks in wood and polymer available and several options for semi-finished custom stocks. It's an ideal platform for DIY home smithing at reasonable prices.
Here's a 45LC/454 Casull Stutzen that I built this Winter. SB2 receiver; 45LC carbine barrel reamed to 454, and a Mannlicher style stock semi-finished from Gunstocks Inc:
And here's a 20 ga that I built on a case color SB1 receiver with Buffalo Classic stock and forend, as pleasant a little grouse gun as you could want:
Certainly worth investigating and brings your Five year time horizoon much closer.
