The localisation long exact sequence in K-theory over an arbitrary base

I'm not sure that what I have to say really addresses the heart of your question, but it seems at least related.

Background

The general Localization Theorem (7.4 of Thomason-Trobaugh) states the following. Suppose $X$ a quasiseparated, quasicompact scheme, suppose $U$ a Zariski open in $X$ such that $U$ is also quasiseparated and quasicompact, and suppose $Z$ the closed complement. Then the following sequence of spectra is a fiber sequence:
$$K^B(Xtextrm{ on }Z)to K^B(X)to K^B(U).$$
Here $K^B$ refers to the Bass nonconnective delooping of algebraic $K$-theory. One thus gets a long exact sequence
$$cdotsto K_n^B(Xtextrm{ on }Z)to K_n^B(X)to K_n^B(U)to K_{n-1}^B(Xtextrm{ on }Z)tocdots$$
(If one tries to work only with the connective version, then the exact sequence ends awkwardly, since $K_0(X)to K_0(U)$ is not in general surjective; indeed, the obstruction to lifting $K_0$-classes from $U$ to $X$ is precisely $K_{-1}(Z)$ by Bass's fundamental theorem.)

The term $K^B(Xtextrm{ on }Z)$ is the Bass delooping of the $K$-theory of the ∞-category of perfect complexes of quasicoherent $mathcal{O}$-modules that are acyclic on $U$. Identifying this fiber term with $K^B(Z)$ is generally a delicate matter. Let me summarize one situation in which it can be done.

Suppose that $X$ admits an ample family of line bundles [Thomason-Trobaugh 2.1.1, SGA VI Exp. II 2.2.3], and suppose that $Z$ admits a subscheme structure such that the inclusion $Zto X$ is a regular immersion (so that the relative cotangent complex $mathbf{L}_{X|Z}$ is $I/I^2[1]$, where $I$ is the ideal of definition), and $Z$ is of codimension $k$ in $d$ in $X$. Then the spectrum $K^B(Xtextrm{ on }Z)$ coincides with a nonconnective delooping of the Quillen $K$-theory of the exact category of pseudocoherent $mathcal{O}_X$-modules of Tor-dimension $leq k$ supported on $Z$. If now $Z$ and $X$ are regular noetherian schemes, then a dévissage argument now permits us to identify $K^B(Xtextrm{ on }Z)$ with $K(Z)$.

Your case

Now I'm assuming that $K(D)$ refers just to the $K$-theory of the ring $k[[t]]$ (and not, for instance, the $K$-theory of the formal scheme $mathrm{Spf}(k[[t]])$), then the discussion above applies to give you your desired localization sequence
$$K^B(X)to K^B(X[[t]])to K^B(X((t)))$$
for any scheme $X$ admitting an ample family of line bundles. If in particular $X$ is regular, then the negative $K$-theory vanishes, and we have a localization sequence
$$K(X)to K(X[[t]])to K(X((t)))$$