Well, scratch anything I've said ... not doable on a 3.6.x ... what amazes me, the 2.2 Assignment still shows in modules .... as (disabled) ... but it had that before me thinks.
Looks like gonna have to install a 2.9.highest version on some box ... dummy down your PHP version then attempt restore of course there so you can get to the Assignment Upgrade Helper site admin link.
Granted a 2.2.9 backup is very old ... kinda wonder about the assignments contained therein .... timeless assignments ... guess they exist, but ...
But ... sent you an EMail .... have access to a staging moodle that was 'frozen' in series 2 of Moodle. It's a 2.7.highest. IF you can send a URL to me where I can acquire the backup via wget can give it a go on that staging server. Convert the assignments there. Make backup and provide you with a link to acquire the converted backup - which would be a 2.7.highest version but one should be able to restore that on higher versions of Moodle.
Lemmie know ...
'SoS', Ken