Hi Damjan.
The problem is not entirely spikes.when on ac supply you get 6vdc minus 0,6 vdc drop(forward bias of the silicon diode) that gives 5,4 vdc.when ac off you get 4,8 vdc(Ni-Mh cell),let’s try it this way, the ac supply charge your battery while the batt supply your project this in line scheme would creat less havocs,save your caps too
Khanh.