To get a close measure of you mast height, take a 50' tape to the top of the mast via a halyard. then pull the tape down to the water at the beam. This will be a bit on the long side....ie if you are really say 30', this measure will be 32-33 or there abouts.........at least you wll be safe with this measure for bridges. If you signal or call the bridge, and they can tell you are low, they will not open, and will radio or say back your ok.
If you do not have a 50' tape, go to fxsails, they probably have you sail measurements. One will be the mast ht from deck to the top for a jib. measure from the deck to the waterline, add a foot, max 2' for additional on the top, and you will again, be pretty close.
Another option if you are lucky, if more than one sailboat is going thru when you do, stay up with them, as they will open the bridges for them as a group, get in the middle, then you know you have it made, whether you need it or not.
Freemont would be my worry. Other than that, with a 23' boat, you are talking 30'ish feet at the most........doing swag. could be 35. My 30'r is 42, and I have a reasonably tall mast.
She drives me boat,
I drives me dinghy!