Rye is commonly used as a "nurse" crop to hold soil following sprigging. I dealt with a problem area today in which a customer had septic work done. Sodding during the winter isn't ideal, so putting down rye to hold it over until late Spring was the solution. I recommend using a perennial rye, which is much prettier and easier to maintain. It is also easier to kill off once the Bermuda fills in (you can use any of the numerous products that contain metsulfuron methyl: Amtide, Manor, etc.).

That being said, before you go down that path, the first question is regarding the bare spots. Bermuda is a hardy grower and is very good at retaining soil, so while you may have some soil displacement following heavy rains, that wouldn't lead to bare spots. Bare spots in Bermuda are usually associated with shade from a nearby tree canopy, or a major issue in the piedmont region of Georgia, poor soil conditions. I would first identify what is causing the bare spots and then form a course of action. If you upload a picture of the bare spots, I would be able to give you a better idea.

For the time being, I will assume that it may have to do with the soil, so I provided a link to an Facebook album I created last Fall for dealing with small areas (300-1000sqft) with fairly minor soil compaction. You could follow the techniques used to incorporate soil amendments before planting rye, and the Bermuda would begin to fill in (assuming there is enough sunlight in the area).

Here is a link that might be useful: Fixing problem areas

Don't do it. Doing so will actual slow down the Bermuda. If this was fall, then the story would be different dues to erosion concerns.

Too close to spring, live with the bare spots for now. Once the Bermuda is actively growing you can rip up sprigs to transplant in the bare spots, or even better go to a garden center and buy a small quantity of sod to fill in the bare spots.

If you use rye now all it will do is rob and slow down the bermuda until the heat kills of the rye grass.

Even if you were going to seed rye, you're 4 months too late.

Like serenity said, you have something else going on there. Bermuda does not waste away from rain. A picture would really help.