'Shooting paper patched bullets means that whatever I bring with me is all that I will be able to get. It'll be my one and only checked bag'
BEWARE: Ammo and ammo components can not be packed in the same case as the firearm so you will need a minimum of two bags. You might get away with it but very likely you will have them confiscated if you can not pack them in a separate locked bag. Also the ammo must be in 'original packaging' - check the regs, they are quite tightly defined.
Re locks: 'TSA' locks are NOT okay for locking cases with firearms, an admission that the keys are all out there! TSA officers often tell one to get TSA locks, unaware that the rules do not permit their use.
Non-TSA locks may well be cut off by TSA (or UK security) when they inspect the contents of your case unless you can stay in close proximity of the case to hand over (and recover) the keys. This is usually required procedure going from USA to UK but often not possible going UK to USA.
If the lock is cut off, technically the gun can not then travel as all airlines have a condition that any case containing a firearm MUST be hard sided and locked.
I always travel to and from the USA with spare locks and spare keys, all of which are totally expendable. One of the joys of travelling with guns!
Lastly, I must stress that the UK have VERY prescriptive rules on what ammunition and caliber of rifle can be used to shoot deer and these vary between England/Wales and Scotland. Generally these rules are carefully observed as any offence against them could bring serious legal repercussions for the hunter AND be a serious home-goal for the hunting community in general. The last thing we need in the UK is to give ammo to the antis.
Your much loved rifle may be perfectly capable of killing deer but completely illegal to use over here. If it uses paper-patched bullets, I doubt it will make the necessary ft/s while the ft/lb may be awesome!
Enjoy your trip!