The logic of Kant's Critique A70-76 in modern form

Consider Kant's table of the 'forms of the understanding'  obtained by abstracting judgments from their content, expounded in A70-76 (B95-101) and divided into quantity, quality, relation and modality.

Judgments are defined inductively.

Let $C_1$ and $C_2$ be concepts. Then $\mathcal{Q}{C}_1ϵC_2$, $\mathcal{Q}{C}_1\overline{ϵ}{C}_2$ and $\mathcal{Q}{C}_1ϵ\overline{C_2}$ are  judgments where $\mathcal{Q}\in \{U,P,S\}$.

Let $J_1$ and $J_2$ be a judgements. Then $J_1\Rightarrow J_2$ is a judgment.

Let $J_1,...,J_n$ be judgements. Then $J_1\vee ...\vee J_n$ is a judgment.

Let $J$ be a judgement. Then $◊J$ and $◻J$ are judgments.

Here $\overline{C}$ is term-negation. As in the infinite judgment 'the soul is not-mortal'.

But Kant distinguishes between judgment and the content of a judgment, in particular the proposition. Also considering Kant's example,  disjunction seems to be term-based:

Let $C_1,...,C_n$ be concepts such that $\bigcup _i{C}_i=\mathcal{U}$ (or rather, their extensions satisfy this) and $C$ a concept. Then $CϵC_1\vee ...\vee CϵC_n$ is a judgment. So we can use Frege's 'vertical line', ancestor of our $⊢$ and reformulate the syntax of the logic as follows:

Judgments are defined inductively.

Let $C_1$ and $C_2$ be concepts. Then $\mathcal{Q}{C}_1ϵC_2$, $\mathcal{Q}{C}_1\overline{ϵ}{C}_2$ and $\mathcal{Q}{C}_1ϵ\overline{C_2}$ are are propositions where $\mathcal{Q}\in \{U,P,S\}$.

Let $P_1$ and $P_2$ be propositions. Then $P_1\Rightarrow P_2$ is proposition.

Let $C_1,...,C_n$ be concepts such that $\bigcup _i{C}_i=\mathcal{U}$ and $C$ a concept. Then $CϵC_1\vee ...\vee CϵC_n$ is a proposition.

Let $P$ be a proposition. Then $◊P$, $⊢P$ and $◻P$ are judgments. 

See also the important remark in B141 where Kant confirms the above presentation. Also B11. Analytic judgments are ${=}_I$ in CIL whilst synthetic judgments are ${=}_N$, the are 'extensional' in Kant's own terminology !

Can Kant even express his 'analogies of experience' - which surely must be judgments ! - in such a logic ? The original version of the second analogy in A was: for everything that happens there is something which succeeds it, according to a rule. Alles, was geschiet (anhebt zu sein) setzt etwas voraus, worauf es nach einer Regel folgt, $\mathrm{\forall }x.Hap(x)\to \mathrm{\exists }y.RegSuc(x,y)$. 

According to Bobzien and Shogry Stoic logic could handle this.