Would there be any problems installing a downstairs radiator fed with
pipes
from the floor above?

It's run from a vented back boiler with CH pump.

Very common. As the rad bleed valve can't vent the pipework, not being at
the highest point, just ensure that all pipework can be vented upwards and
you're fine. Unless badly designed, the preexisting pipework should already
be vented, either via the boiler vent, or through manual or automatic bleed
valves. If you are adding new pipework, ensure that there is some venting
mechanism in place, i.e. any air will rise to a bleed valve or open pipe
end.

Christian.