The difficulty is caused entirely by the decision to use algebraic (K)-theory of finite–dimensional algebras.
If one replaces “algebraic” by “topological” and works with (real, (\Bbb Z/2)-graded) (\mathrm C^{*})-algebras the whole picture becomes canonical and all the problems disappear.

1. the (K)-theory of a point is the non-connective 8-periodic spectrum (KO);

2. Morita–invertibility in the graded world is the same as invertibility under the graded spatial tensor product, so the Clifford algebras already form the Picard subgroupoid of the 2-category of real graded (\mathrm C^{*})-algebras;

3. there is a symmetric monoidal functor
   [ \mathbb K\colon(\mathrm{C^}{\Bbb R}^{\mathrm{gr}},, \widehat\otimes)\longrightarrow (\mathrm{Sp}{KO}^{\times},, \wedge_{KO}) ] whose value at a graded (\mathrm C^{})-algebra (A) is a (KO)-module spectrum (\mathbb K(A)) with
   (\pi_i\mathbb K(A)=KO_{-i}(A))
   (real topological (K)-theory of (A)). The construction is due to J. Joachim (Mem. AMS 2003, Th. 5.27; see also Joachim–Johnson, Topology 2005) and is simply [ \mathbb K(A)n :=\mathrm{Hom}{\mathrm{gr}\text{-}C^*}(,C\ell(n,0),, A\widehat\otimes\mathcal K), ] with structure maps given by the Bott class.
   It is lax symmetric monoidal; after restricting to the Picard subgroupoid it is symmetric monoidal.

4. Because (C\ell(p,q)) is Morita–equivalent to (C\ell(1,0)^{\widehat\otimes,p}\widehat\otimes C\ell(0,1)^{\widehat\otimes,q}) and
   (\mathbb K(C\ell(1,0))\cong\Sigma KO,; \mathbb K(C\ell(0,1))\cong\Sigma^{-1}KO) (Bott periodicity), we get [ \mathbb K\bigl(C\ell(p,q)\bigr);\simeq;\Sigma^{,p-q}KO \quad\text{as }KO\text{-module spectra.} ]

Hence

Clifford algebras  ——( 𝕂 )——►  invertible KO–module spectra

is the required symmetric monoidal functor; on (\pi_0) it is the Atiyah–Bott–Shapiro isomorphism
[ \mathrm{Br}_{\Bbb R}^{\mathrm{gr}} ;\cong;\Bbb Z/8 \xrightarrow{;;\simeq;;} \mathrm{Pic}(KO);\cong;\Bbb Z/8 . ]

Why the two “ways of killing trivial objects’’ disappeared

• With algebraic (K)-theory you first get the connective spectrum (ko) and have to impose periodicity by hand, hence the choice between “square = +1’’ and “square = −1’’.
• In (\mathrm C^{*})-algebra (K)-theory Bott periodicity is built in; the two generator algebras (C\ell(1,0)) and (C\ell(0,1)) are already inverse to each other in the Picard groupoid, so there is nothing to choose.

What about a general ground field (k)?

Replace (KO) by the 8-periodic hermitian (K)-theory (Grothendieck–Witt) spectrum (GW(k)[\beta^{-1}]) obtained from the algebraic spectrum (GW(k)) by inverting the Bott element (\beta).
There is a completely parallel functor

graded central simple k-algebras  ——►  Pic(GW(k)[β^{-1}])

sending the graded Brauer group of (k) to (\mathrm{Pic}\bigl(GW(k)[\beta^{-1}]\bigr)); see Karoubi “Algèbres de Clifford et (K)-théorie hermitienne’’ (1978) and Schlichting (“Hermitian (K)-theory of exact categories’’, 2010). Again no choice of a quadratic character is needed, because inverting (\beta) identifies the two classes.

Summary

Use topological (real) (K)-theory of graded (\mathrm C^{*})-algebras instead of algebraic (K)-theory of finite-dimensional algebras; the Joachim functor gives the desired symmetric monoidal map, produces (KO)- (not (ko)-) module spectra, and makes the graded Brauer group and (\mathrm{Pic}(KO)) naturally identical.