I have a deck and arbor business and I first never use the 5/4 boards. Have yours began to warp and twist and become unsightly yet?
Think of this, the dead weight on that area. You probably will need 450 gallons of water at 7.5 pounds per gallon, then you have your structure, then you have 4 people. Thats alot of weight on that one small area.
Hot tubs assembled on existing decks usually does not work out at all. The proper thing is to encorporate the hot tub in the intial design and construction of the deck.
My opinion, is to remove the 5/4 decking in the hot tub area. Double up on your beams, then double up on your joists spacing, then in the area of the tub, install 2 x 6's or 2 x 8's for the decking. Continue the 5/4 everywhere else. You can actually conceal the 2x under the spa.
I also do not like the idea that the deck is attached to the house via ledger NOW with this hot tub. I would like for the deck to be free standing and any settling would not pull the housing down with it.
Consider my weight thinking and just how much that one small area will carry. your posts holding the beams could sink.
Another thing to consider. are the posts the beams are attached to sitting on top of the posts or fastened with screws or such. If the beams are fastened on the side to the posts, they will eventually break. The beams need to sit on the posts. Are the posts sitting on a concrete footing? If they are just in the ground and concreted in, they will sink.
You have many issues to consider and it would be not good to have to take this down after installation and re do the deck. easier to do it NOW.