When bermuda is mowed down as low as it should be, there is not much you can do to stop the weeds from looking weedy. If you don't get rid of all of them with RoundUp, you'll have them back.

It is unlikely you need topsoil for any part of this renovation. Unless you have low spots where there is standing water for days after a long rainstorm, then you don't need more topsoil.

Here are my suggestion for a tune up.

Step 0: On April 15th, fertilize with alfalfa pellets
Step 1: On May 15th, Roundup entire lawn
Step 2: Water briefly, 3x per day for 5 days and roundup again
Step 3: Water briefly 3x per day and RU again
Step 4: Wait 5 days then scalp and remove the clippings.
Step 5: There is no step 5 in this new context.
Step 6: Spread seed
Step 7: If you feel a strong need to cover the seed, use compost
Step 8: Roller over the seed and/or compost
Step 9: Fertilizers only work on roots. You don't have any yet. By the time you get some, the started will be somewhat depleted. The reason for the alfalfa up front was so you would not need starter fert.
Step 10: Water 3-4 times daily for a few weeks

Notes:
If you start this process now, you will waste all that time, effort, and money because bermuda seed needs hot soil to germinate. The approach I outlined will get your seed on the ground in June. That is perfect timing.

Alfalfa pellets are an organic fertilizer. Organics are a biological process whereby the microbes in the soil convert the pellets into plant food. The process takes 3 weeks and does not deplete like chemical fertilizers.

If you scalp off the tops before spraying with round up, then there is nothing for the round up to attach to. It needs to land on the blades of grass and weeds to take up the round up into the plant. Don't scalp.

Definitely roll the new seed to ensure good seed-to-soil contact.

Topsoil is unnecessary for any of this. Mother Nature seeds by knocking seeds on the ground followed by herbivores stepping on the seed. Your soil will be very moist with the weeks of watering.

@dchall_san_antonio Do you have any tips for fertilizing with the alfalfa pellets?

I put some down last year, watered them, they would expand and "explode" but they just seemed to sit on top of the grass. Is there a more effective way to ensure they reach the soil?

After the pellets explode, drag your hose across the grass to knock the pellet worms off the top and down into the grass. You can drag anything but the hose gets it all in one pass.