I do not have any experience with Encores, but I do and have owned several Handi rifles. I too reload and have several of my resources tied up in the 44 mag specifically for deer hunting. Great cartridge. The handi's are great rifles, but can, not always, be a project to get them to shoot good and even great. All the DIY fixes are outlined in the Handi FAQ secion on this forum and are relatively easy to accomplish. I like the lego aspect of the Handi system and do a lot of my own customizations. Doing this on a under 300.00 rifle system is a little easier on the ole conscience than a more expensive platform, especially if I bubba it instead of customize it. Been known to happen sadly. I enjoy the tweaking to get the accuracy, so the work I put into one is a plus for me. I free float and bed all my action/barrels. Most if not all have custom stocks,bed the scope rails so it will never, ever move, etc...
So I guess it is a toss up in my book. Easy way, get the the Encore barrel. vs The maybe not so easy, but can be equally rewarding, depending on your DIY skills, and getting to explore another rifle platform for about the same amount of money Handi rifle. The latter would win me over personally.