I’d allow the rootstock shoot to grow..  If any (assuming there’s more than 2 or 3 Chocolate buds) of the grafted scion buds begin to break and leaf - I’d immediately prune the root shoot back to 2 or 3 sets of leaves, removing it’s terminal bud and top growth.  

Persimmons grafted while dormant are extremely slow to emerge … I once assumed 3 I’d grafted had died ..until some time in mid-July when they all broke buds and grew ..into strong productive trees (Jiro’s).  

I’ve 3 massive pots sitting 8 feet from me right now, all whip & tongue grafted the same time as yours (only 3K away, by me), Am. pers. to (better) American persimmons: Two show no sign of bud break (stock or scions); one has both it’s 2 scion buds open - while two rootstock shoots ‘just’ got their terminal leads removed, to force all energy to the grafted emerging buds up top, and allow the root shoots to produce a limited amount of energy…  

The rare but special times I’d graft persimmon at the HOS event I’d warn their owners to ‘be patient,’ if the graft was clean & tight, it should only take time ~

First, I agree with John’s recommendation to keep up on watering.

Second, as it’s a week later - of my 3 newly grafted persimmons, the first to leaf out has two - 5 inch lengths of scion growth, I'll tip-pinch one soon...  The two (pinched) leafed rootstock buds are simply feeding into the union, and not ‘their own growth.’  

Of my 2 remaining grafts, a second has opening buds … while the third is still tight…  Fingers crossed, but it can take time 

I’d tip-pinch two of the three rootstock shoots then stake & protect the strongest.  You want to send the bulk of the energy into one ‘trunk.’ I’m not sure if ‘Budding’ will be an option (in two months) with such new & tender growth, but I'd make another graft attempt if this one doesn't kick in…   

I’ve unwrapped the dissolving grafting bands on my 3 persimmons; 2 have calloused over and are growing well, a 3rd shows no callusing or shoots from it’s rootstock … which leads me to believe, so far, the rootstock was bad ..as the graft was clean & tight…  

I do not consider whip & tongue dormant grafting the best way to propagate persimmons.  Budding, as with most ‘whip’ fruit trees, is best.. You may look into that for the following season (over a year from now, Aug. of 2019), as I’m sure you can do better than fruit from the rootstock.  The HOS Arboretum may have ‘Bud wood’ available; procuring the source wood is often what makes Budding most difficult.