Good question...I'm not sure exactly how that would work, since I've only recently started using stablized burl and buy it already treated and dyed. If there is a business or individual willing to stabilize the wood for you, it may be best to go ahead and cut the blocks out yourself, or request that the burl be cut to specific sized blocks prior to stabilization. The reason is that even in a high quality vacuum chamber, the wood stabilizer and any dyes can only penetrate so far into the wood. If stabilized whole, the center might (note that I said "might"; not "will") still be left untreated. I'm new to using burl wood, so take this with a grain of salt.
P.S. A lot of it will depend on how wide or thick the blocks are. Length isn't going to be too much of an issue (I wouldn't think so, anyway). 2 inches is the widest I've ever seen in stabilized wood.