Can the manual transmission be repaired? Can you change its oil? on 2006 Mini Cooper

I have a 2006 manual transmission Mini Cooper S. At 60,000 miles the third gear sinchronizer started to fail and I los 3rd gear. The dealer mechanics say that they can only replace tge whole transmission at roughly 8K, and that they cam't even replace the transmission oil.

Asked by for the 2006 Mini Cooper