TOS-R sort of puts this to rest by explicitly showing how the Botany Bay
is abandoned in space, cargo containers (or whatever those five remaining wedges out of an apparent maximum of 16 are - fuel tanks, perhaps?) and all, long before the action moves to Ceti Alpha.
Kirk probably wouldn't bother to go back to that ship just to get Khan some survival goods that the Enterprise
could easily provide herself. And Khan doesn't appear to have all that much stuff down on the planet anyway. Quite possibly, just a few crates of Khan's personal belongings, books and the like, were beamed from Botany Bay
while the two ships were flying in formation, either by Kirk or then later by Khan. And perhaps a few of those were strapped together with belts containing the originating ship's name.
Alternately, those containers being hauled around by workbees in ST:TMP are in fact an old model, its standardization going back all the way to the 1990s, and thus the Botany Bay
happened to have some aboard, too. Many of the cargo straps inside the containers seem to serve no function other than hanging from appropriate fixtures - sort of suggesting they are scenery coming with the rusty walls, rather than vital items carefully removed from the Botany Bay
and installed here for a purpose.
The size issue is IMHO easily fudged. The containers offer standing height here:
And the ones in which Khan lives are obviously "double-enders", with two curved walls, as per the above picture featuring the wall stenciling. If the interior height is still a tad too much to be plausible (it's about twice the ST:TMP height!), we can argue that Khan removed the bottoms of the containers and placed the tops on solid stone foundations, and the floor we (fail to) see is actually in a pit.
Or then the containers simply come in a variety of sizes. Added height wouldn't inconvenience the workbees hauling these things, nor would it affect the stacking of the containers on the floor of a starship cargo hold, or on assorted other flatbeds. It would simply make the higher containers incompatible with the fancy wall sockets, but that may be of little concern overall.
Also, at least some bits of the containers did explicitly come from Starfleet, as per this pic...
But the piece Khan is using as a door here need not necessarily be related to the containers themselves; apparently, the piece is a later add-on.