The total power of the battery is :
P = V I = 12x10 = 120 watts
Since you want to use the car battery for a max of 3 days, and use the battery for 5 minutes every 35 minutes, then you will switch on and off the battery for:
Number of on/off = 3x24x60/35 = 123 approximately
Total time the battery is being used is t:
t = 5 x 123 = 615 minutes = 10 hours approximately
Therefore, the total consumption of energy (W) in 3 days is:
W = Px t = 120 x 10 = 1200 watt-hour = 1.2 kilowattt-hour
Suppose that the cost of each kilowat-hour is 50 cent
Then cost of this amount of energy is = 60 cents
Which I assume is within the budget. So go ahead and connect your car battery to your electric device the way you want..