Dear David,
I can reproduce what you describe. I also suspect that it is probably an unwanted feature, i.e. a bug. Other formats in ExportString seem to work ok-ish.
ExportString[(1 + a)/b, "TeX"]
seems to work alright in the cloud. Importing it:
ImportString[%]
gives

The error messages say
General::stop: Further output of FrontEndObject::notavail will be
suppressed during this calculation.
FrontEndObject::notavail: A front
end is not available; certain operations require a front end.
FrontEndObject::notavail: A front end is not available; certain
operations require a front end.
FrontEndObject::notavail: A front end
is not available; certain operations require a front end.
Ops, this is strange. I just re-ran your commands in the cloud and now they give this:

which is the output I get on a Desktop as well.
Either WRI are working on this right now or I do not understand this behaviour at all.
Cheers,
Marco