While Emma might be right ... think the number of courses might be a factor ... in which case ... some 2 cent thoughts.
Interesting 'project' ... first, it is doable but not via web browser and some research needs to be done on the differences (cusomizatons/addons/themes/mods/blocks/etc) between the 2.3.x and the 3.3.x and addressed prior to beginning the 'project'. Example: a custom theme won't survive the leap and if courses were set to that custom theme (contained in the backup of the courses) then upon arrival, reset of theme and maybe some other items would need cleaning up .... like links to files that were aliased (don't re-call if 2.3.x had that feature or not) and other things ... like a addon mod/block that existed in 2.3.x but not available in 3.3.x.
Did something similar from 2.6.x to a 3.3.x .... all from command line ... and both instances were on same server (made it easier for sure, but is doable if the instances are on different servers ... just takes a little longer).
Are the servers involved Linux? and do you have ssh/command line access to both?
'spirit of sharing', Ken