I have been tasked with creating a +3 magic item (glove) that controls a 15 foot long monofilament whip. It is supposed to count as a monk weapon, utilizing the monks martial arts die (currently a d8), the use of DEX modifier, and proficiency bonus. I think I could get the general result desired if I could model it after a dagger +3, if I could trade the thrown range for a 15 foot reach. I have been trying to figure out how to do that, but it is beyond me. Any ideas on how to accomplish this? Another approach would be to base it on a (martial arts die) rope dart with a 15 foot reach, but I can't figure out the programming for that, either.

It’s not possible. Reach is a specific weapon property that only adds 5 feet, and the base range of the dagger cannot be altered.

The martial arts die overrides the damage die of the weapon if it is higher so you don't have to change the die on the weapon.

As said above, Reach only adds 5 feet. You should just put the ability in plain text even though it will not show on the character table.

Personally I would use a handaxe which does slashing damage - matches better than piercing. It also has 20/60 range built in and in the text you can say it can only go 20 feet no long range. /shrug it's a very rare class weapon if it has +3 so I would probably just give it the regular ranges anyway.