Nice one.
The carb should not flood. The float valve should shut off the supply if the chamber is full - the pump should just idle, so it looks like the carb will be in need of repair or replacement at some point, unless the pump is the wrong one? The 500K had a different pump to the various models used on the 500 and 126 as the K engine is pretty much gravity fed apart from when going down hill.
I can't find a part number for the K pump - there were at least 4 for the 500F and probably many more for the 126.
Axel Gerstyl do a pump (18 euro) for the K, they also do the bakerlite spacer etc. They do a repair kit for your carb for 21 euro.

the 500/126 center (Fiatclassic.com) have a set for 15 euro or a carb for 131 euro (ouch).
