I had a similar problem when my Mk3 Jazz was indicating low battery in the key fob after a ridiculously short time like a couple of months after putting in a new battery. I tested the supposedly exhausted battery and it was good, round about 3V. I cleaned the contacts in the battery compartment with a cotton bud dipped in isopropyl alcohol, although I couldn't really see any corrosion or dirt. I put the old battery back in, the low battery message went away and the key fob now seems to be using batteries at a normal rate.