Dear Peter, I will answer to the question in the comment since it seems this is you main interest. For the algebraicity of the $p^n$-torsion points of the universal deformation, I gave a proof of this to Matthias Strauch a few years ago. He included it in his article "Deformation spaces of one-dimensional formal groups and their cohomology", this is theorem 2.3.1 (the proof as written in Matthias article is for Lubin-Tate spaces but works in general without changing anything), see his webpage. You don't need deformation theory for finite flat group schemes for this...look at the proof there's a trick (due to Artin).

For Brian, you say "Do you mean there's a BT-group over a finitely presented algebra whose pullback to completion at some point is the universal formal deformation? If so, then I find that hard to believe". But in fact this is conjecturally true ! This would follow from the non-emptiness of Newton strata in unitary PEL type Shimura varieties at a split prime $p$. For example this is known for the deformation space of a principally polarized BT group thanks to the non-emptiness of Newton strata of Siegel modular varieties (I mean you deform not the BT group but the BT group together with its principal polarization).