"What do yall think would take the cake? Specifically referring to target/sniper rifles built from a military issued battle rifle..".
IMO, the Swedish Model of 1896 has the most impressive history of a standard military bolt action rifle that was adopted at the very end of the 19th century, then became a sniper rifle during WWII, and in the post-war period was further developed into 3 very formidable target rifles, especially suited for 300m competitions., where they were very competitive. The brief history is M1896 standard infantry rifle then M41 & M41B sniper rifles, then M1896 FSR (target sights) followed by the CG-63 (dedicated 300 meter target rifle) followed by the CG-80 (1980) and finally the FSR 89 (1989).
The standard M/41 Swedish 6.5x55 military ammo is considered by many to be basically 'sniper grade' ammo, which certainly helped the reputation of the M96 model rifle and all its variants. It has been said that the M41B with the standard issue M/41 ammo can be a 1-MOA rifle, and thus perhaps the most accurate sniper rifle of the WWII era. I can not validate that claim, but I have read that Sweden tested their M1896s and only the most accurate rifles were selected to be converted into sniper rifles during WWII. All of them were rebuilt/updated in the 1950s and continued their service until I think about 1990.
Here's a 10 minute video on the history of the Swedish FSR (volunteer marksman) and the rifles they used, with particular focus on the FSR 89, the last target version of the Swedish Model of 1896. The FSR 89 represents almost 100 years of accuracy refinements based on the old M1896 long rifle....
I have an M41B sniper rifle (built on a 1900 dated action), and a nice CG-63, both are quite accurate. (I briefly had a CG-80 but never shot it, as a serious collector of Swedish rifles talked me out of it - a decision I regret as I have not been able to replace it..) Here's an article on the M41/B, which the author thinks is tied with the British Mk 4 (T) as the two best sniper rifles of the WWII era.
http://www.gunsandammo.com/historical/sniper-supreme-model-41b-swedish-mauser/
Attached are some pics of the CG-63, CG-80 (both early and late versions) and a rare FSR 89 owned by the collector who bought my CG-80 (the rifle with the stainless fluted barrel is a customized CG-80, where the FSR 89 is the rifle at the top in those pics). These are not my rifles, but to the original posters question, the Swedish M1896 gets my vote for the "baddest military action - sniper/target rifle" of the 20th century.
The Springfield 1903 comes in 2nd place, but the development of that rifle as a sniper or target rifle effectively ended with the conclusion of WWII, and I guess the M1891 Mosin-Nagant gets an honorable mention as it was also used in Finland as a target rifle and sniper rifle in the post-war period - but the Swedes kept perfecting their 1896 Mauser-based rifles in the four decades after WWII had ended, and I can't think of any rifle platform that outshines the interesting history of the Swedish Mauser and its sniper and target rifle variants...