If the rifle has been sporterized, do what ever you want. IF the rifle is still in GI issue configuration, ANY alteration is potentially throwing money away if you think you might ever sell the gun.
Any permanent modification lowers or destroys the collector value. Its your gun, do what you want.
Excellent replacement triggers for the 1903 were made by Timney, Canjar, and a lot of other trigger makers over the years. There are probably some still on the market, though there is little demand these days.
The advantage to a replacement trigger is that it can be undone and the rifle returned to original condition (provided you KEEP the parts).
The downside is the cost. Both ways. Cost of buying the trigger, and tht cost in value the new replacement trigger removes from the rifle.
50 years ago, the situation was the exact opposite. GOOD QUALTIY modifications (sportizing) ADDED to the value of the rifle, because the buying public wanted a good looking, good handling rifle to USE, more than they wanted a pristine GI issue condition milsurp to collect.
I've got a beautiful 1903 sporter, masterfully redone, rebarreled to .25-06, restocked, scope mounted, reblued & polished, 32 oz trigger on my scale. Lovely rifle. Shoots pretty good, too!
That gun, even with a scope on it isn't worth much on today's market. Not even half what a collector would pay for the very same gun still in GI configuration. Its a crazy world nowdays...probably always was, just seems moreso lately, but, that could just be me....