I have used several approaches to Kudzu Control. The big stuff back in the woods I just cut off, some as big as 2" in diameter going up into the top of the trees. One hill side that was pretty clear of trees but solid Kudzu I spread some old tarps over and weighted them down and let the Kudzu bake for a couple of years with no sun and it looked like it worked then I decided to extend a flat area and dumped 5 very large dump trucks of shale on it, that seemed to work so far. Areas I can mow works till I stop mowing. I found the Coop generic brand of Trycera herbicide works well keeping it from spreading, same stuff the power company uses but I dilute more than they do.
I tried to get a local goat farm to put a few goats in one area but since it had water running through it, he was afraid they would drown themselves. I paid a guy for 40+ hours of work with an excavator and 9 normal dump truck loads of "clean" fill to make that area so I could mow it once a month and get to another area to spray.
To get back to clover I think the nitrogen rich fertilizer and more grass seed is probably the best bet. Since clover puts nitrogen in the ground and cause the grass to grow then gets crowded out till the grass needs more nitrogen, if the nitrogen was supplied by fertilizer the grass should keep growing and crowding other things out. If you are not fussy what kind of grass is in your yard you might try adding some fast growing variety like rye grass.