Davo, the fan only draws 150/170 mA, while your t'scope may draw 2 or 3 amps when slewing. Not sure when tracking,(~200/300mA???).
The peltier requires a constant minimum 4 AMPS to work 'efficiently'. (if they do at all)!
IMHO, You would be better off buying the biggest capacity battery you can to start with. You will not get a true test result using the 7 AH. T'will 'die' within 15/30? minutes. Put a DMM across the battery terminals & watch the voltage. If it falls below ~10.5 volts, stop. They don't like going below that.
HTH.
L.