My 20+ years in GA taught me also that sheetmulching is the only way to manage bermudagrass (and it's sister scourge, nutsedge). Sheetmulching every year, that is, preferably laying paper and cardboard over the grass as it's growing (rather than when it's dormant, giving the paper a chance to soften first) Transplanting sizeable plants into this, or simply leaving mulched areas for a season to subdue it a bit first, is also adviseable. Small direct seeded crops are the most difficult.
But a plant the size of a mature blueberry bush ought to have no trouble surviving in a bermudagrass sod, especially if it has adequate
irrigation. Sweet potatoes, as stated above, and also winter squash, make good groundcovers. To be at it's most vigorous, bermuda and nutsedge need full sun, so overshadowing them with other vigorous plants is helpful.
Another lesson learned the hard way is not to have any kind of permanently edged raised beds, paved pathways, and so on in an infested area. The roots will get up under and beside these and then creep out on top of your sheetmulch. You want to be able to paper the whole area, completely, and the better overlapped it is, the better and longer the control.