nothing wrong with the 6.5x55, and likely to be easier to feed as you say, Bomag has a bit of experience with long range .243AI for target might be worth a PM, .308 is not overkill for small deer, and with the right bullet will do a surprisingly good job with little meat damage compared to some of the higher speed smaller calibres. The .223 is an ideal fox round and by all accounts with the right bullet will do the business on small deer for less powder then the .243 or 6.5x55.
don't worry too much about buying Lee stuff, it will load accurate ammo and thats all you need at the moment, if you later decide to go more up-market then you can always sell on the Lee, and it will cost you a hell of a lot less than some of the other brands (at this point you don't even know if you will continue reloading) somethings are worth getting , such as a balance scale, the Lee will work but is fiddly see what you can pick up second hand, the Lee length cutters are calibre specific and for .223,.243,.308,6.5x55 et al will be more than adequate, no need to break the bank at this stage, calipers can be bought very cheaply now, although there are always people who will advise you spend £200, although how many of them also spend on recalibration I don't know
if you really want to just dip your feet then I would get a Lee hand press, they are awesome little devices and will be of use for years to come, I know some very finnicky reloaders who use them on the range to fine tune their rounds (you can use almost any make of die in them) none of the calibres you are looking at would cause a problem even full length resizing is easily done with them!
PS we take anyone in the BBS, you only have to look at the membership to see what I mean !