FH is very hard on roots. University trials on fairway bermuda show negative effects on fill in and recovery if used on turf that is not totally established at time of application. From what I have seen, if grass is root pruned with something like pendimethalin or any other orange herbicide, those nodes affected are permanently damaged. New roots have to come from new nodes. I would look at using the rate of Barricade labeled for use over newly sprigged bermuda if grassy weeds are a concern. Gallery is useful for broadleaf weed control.
Of course, there is nothing wrong with keeping the area mowed low until the plugs have achieved >75% coverage and doing what you have to do. Starting low mowing soon during the grow in tends to make weed control much easier. Many weeds seen during the grow in are not normally a problem in bermuda mowed low.
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