Thank you,

As far as growing a mix, I'll pass on this question cuz I'm not familiar with Bermuda grass and the types of that.
I'm like to use a combo of grass types for our northern zone. Forty percent Kentucky Blue, same percentage perennial ryegrass and 20% fescue. Each gives its best in sun/shade/drought and winter cold.
Your Bermuda is a good one for shade as well as sun but is more proven when shade is more prominent.

Why not with the top dressing, do an overseeding of a type that meets your needs.
As far as what kind of topdressing, compost (if its good compost) is a good choice. Triple mix or good topsoil can be mixed with peat moss and/or sand to improve drainage.
One half to one inch is usually all that is recommended as a topdressing. More would be given to the depressions and sometimes, rolled to level out the surface.
If you do roll, be sure to add no more than 1/3 the water volume or you will have a time moving it. Less than this amount can often do a good job.

As for what to use depends a bit on what is available to you. You can use compost, but you mentioned you are doing this to level and smooth out the grade, and compost is worthless for that. Compost will just decompose into nothing.

As a pro I like to use what is known as Pro-Mix. It is 50/50 by volume of a real coarse sand and PEAT, not peat moss.

Really coarse sand works great, or a good quality top soil works great too.

I am in zone 5a. I usually do it around fall because it rains more often at this time and helps wash in the compost.

Paul