Salting the asparagus plants' bed is an old-timey way to kill other weeds without hurting the asparagus because it is very salt-tolerant. I've never used this method and don't know if it would be effective on bermuda grass. This old-timey method fell from favor after the development of herbicides, but historically was used prior to that. Modern research indicates salting the soil can help combat some soil-borne diseases like fusarium crown rot. You'd have to Google to find the amount of salt needed. This probably wouldn't be my first method of choice because my soil already is high in sodium, but since it is a recognized method for controlling weeds in asparagus, I did want to mention it.

I just dig up bermuda when it first appears in the asparagus bed and is easy to eradicate and never have left it long enough to let it become entwined in the asparagus roots. I don't necessarily think that digging up and resetting your asparagus roots would put you back at the equivalent of Day 1, but I wouldn't harvest any asparagus spears in the same year I did the digging. Once you get the bermuda grass out, mulch heavily to help keep it out. For many people, repeated digging and heavy mulching have helped them get the bermuda grass out of the asparagus beds without digging up the asparagus roots, but it might take 2 or 3 years of repeatedly digging out the grass to finally get it all.

There are herbicides labeled for weed control in asparagus beds but I don't use any of them so don't know what ones they are. I think that an old one called Poast could be used in asparagus, and I assume it is still on the market but don't know that for sure. A herbicide of any kind would be my last choice after I had failed at digging and mulching.