Update:
The LS 130 was a great investment, and I'm finding all sorts of ways to use it that I didn't anticipate.
The V grooves have become a two step process. I tried to attack them straight away with the LS but the old finish on cedar gummed up the paper far too quickly for it to be an efficient one step procedure. I drag a scraper down the groove to clear it out, then all that remains is a slightly waxy haze on the wood. One pass with the LS is all that's required to clean that up.
After a few passes I do start to get some gumming up on the paper which can seriously shorten the abrasive's lifespan if not quickly remedied. I keep an old, soft bristled toothbrush in my pocket and, with the sander still running, run a brush along the corner to remove the dust. Doing this occasionally prevents any buildup from forming a waxy film on the abrasive.
The client is amazed that the V grooves can possibly be preserved so well, so all is well in my world.
Thanks for all the replies!