There's no right answer....I've sporterized a large number of WWII rifles.....so here's my advice:

The 1903-A3 is a .30-06 and certainly worthy of deer and hogs..... However any attempt to sporterize it will only reduce the value.....Unless you're darn good at it.

I'm assuming you're a younger hunter....Put that Springfield 03-A3 in the closet and drag it out twenty years from now.....You'll see it in a much better light!...Don't sell it.....don't sporterize it.....shoot it but as you say...It's a bit heavy

Now that .303 British Lee Enfield will also kill deer and hogs and IMO will never be worth much.....(some will disagree.....but they're Brits and Canadians.....let them get their own Enfields!)

You can take a hack saw and a rasp and any other tools you want to that rifle.....It will never be as good as when you're done and I hope you cut off everything that reduces weight without injuring its strength! It's a fairly powerful rifle and strong enough for the .303 British cartridge it was chambered for.

Remember that no matter how nice a job you do it'll still be darn near worthless.....except to you!

Now.....here's the best option.....put the Springfield in the closet.....leave it there for at least twenty years.....and then look at again to see if you want to have a talented "smith" turn it into a great rifle.....

Sell the Lee-Enfield and take that money and add a few bucks with it and cruise the used gun racks looking for a Rem -700 or a M-70 push feed for about $400.....you now have a real fine gun to hunt with and another waiting in the wings.....

well...you asked!