I use a bucket primary for my sour beers and use it according to the beer I intend on getting. If I'm brewing a sour beer with the intentions of having a scant amount of acetic quality then I might leave it for many weeks without disturbing or opening; if I do not intend on any acetic then I'll simply perform the initial (sacc) fermentation and then move to secondary for various amounts of time. If I'm doing a very long-term spontaneous then I'll rack over as much as I can (yeast, trub, and all), whereas if I intend on a shorter or less funky secondary aging then I'll leave behind what has settled after 10 days day of primary knowing that most of the microbes will still be in suspension. Secondary (a highly oxygen-resistant vessel) is always filled to within an inch or two of the bottom of the stopper/airlock/etc.
As for your situation with a 1.001 gravity beer after primary sacc fermentation:
#1 with BrettC. Most likely, given time, this one will develop some typical brettC characteristics given some time without much intervention provided you minimize oxygen pickup during the secondary aging.
#2 with LactoB. My guess is that this one will not do much during it's secondary aging, or will not do it very fast. There will likely be some souring over time but how long is a total guess.
In both cases, if you want to move things along a little quicker, malto-dextrin powder (i.e. carapils being the grain form) will provide both microbes (lacto and brett) with a food source on which to work under the mostly-attenuated environment they're already in, and is easy and inexpensive to provide. It would be boiled and cooled in a small quantity of water first before being added for sanitation purposes, of course. As is the case with anything related to brewing, there's no guarantee of anything and someone else will recommend against doing it
At the end of the day, you can wait it out and see what happens. If you find nothing happening, or not to your expectations, then down the road you can try to intervene a little bit. I had to do something very similar with a sour saison that came to <1.000 FG with brett and pedio as my long-term microbes. After many months of almost no change, I intervened with a sugar source that pedio could work on that was not a source for brett and the result was/is fantastic.