Terry,
My thought would be that if you don't trust that strong back your probobly right its gonna give you problems. You can stiffed and brace it, but ....
Its all gonna end up being a gamble taken, that it will work across the life of your boat building projects.

If your going to stick with what you have I would consider sheathing the whole thing in something stronger, but even that is something of a hail Marry pass.

My thought would be if its floppy you might end up putting a ton of good $ and time and resources and be unhappy with the outcome. Not worth it.

I am the admited king of overbuilding. However, with that said. My suggestion would be trust your gut. Ditch old floppy, and start over. Don't look at it as a waste but instead as another BB lesson learned. Head out the lumber yard and get something stiffer and thicker and build the strong back you really want.
Consider it as an investment in your next several boat building projects.
Enjoy
!RUSS