Your hot tub measures 7x7x3. To be on the safe side, let's assume that this is the actual mass of the water it will contain and nothing else.
147 cubic feet of water
7.5 gallons in a cubic foot
147 x 7.5 = 1102 gallons of water
Water weighs @8 pounds per gallon, so your hot tub will weigh some where in the neighborhood of 8000 pounds. Your hot tub has a foot print of 49 square feet, spread over 12 joists.
8000/49 = 164 pounds per square foot.
Now, we could say that your Hot Tub really only has the capacity of 6x6x2 feet, or 72 cubic feet. You are still looking at about 500 gallons and 4000 pounds.
I will bet that the deck will only reliably support about 60 pounds per square foot, and less than 2000 pounds aggregate.
You don't say how high the deck is off the ground. This makes a HUGE difference as to whether even a 6x6 is big enough. Quite often a spa will have a built up concrete platform that it sits on and then the deck is built around it.
I would have the spa company make some recommendations in your situation. I can't see your deck from here.