What Kate said...
Round up is designed to be translocated through the photosynthesis process. If used at the labeled rates (1-2%), it is near impossible to find in the soil after application. If the tree is in decline, just make sure you DOCUMENT exactally what you are doing--don't want Round-up to get blamed for something it did not do. If you dump in down at 40%, you might cause some problems, but I'm not even sure of that. 40% does work on cut stumps.
You could also lay down black plastic on a hot sunny day. But that could cook some tree roots too...
If you insist on ripping up the turf, a sod cutter would probably pay for itself in time unless you have nothing else to do... You are doing damage to the roots by shoveling or sod cutting though
You coud also AirSpade the area to tear up the grass with that AND improve airflow in the soil all at once. I have never used a sod cutter, but I'm thinking an AirSpade will not be much slower, and you get to leave the stuff there rather than trying to figure out where to put the sod. Probably will not give you a very effective kill though. Round-up plus AirKnife would...