I asked this a while back and got no replies. I think I have a proof now but I'm a bit suspicious of how easy it is (or perhaps I was overthinking it from the start). In any case, I would appreciate some verification.
Let $\mathbb{B}$ be the Borel algebra of borel mod meagre sets in $\omega^\omega$ (i.e, $A \sim B$ if $A \Delta B$ is meagre where $A, B$ are Borel). I want to show this is complete, i.e, every subset $A \subseteq \mathbb{B}$ has a unique upper bound $\sum A$. Let $[B] \in \mathbb{B}$ denote the equivalence class of the Borel set $B$ with respect to the equiv relation above.
First, $\mathbb{B}$ has a countable dense set: $\{ [N_s] : s \in \omega^{<\omega}\}$. From this one can conclude that $\mathbb{B}$ has the ccc- this follows from countable dense subset fairly easily.
Now suppose $[B_\alpha]_{\alpha < \mu}$ be a sequence in $\mathbb{B}$. To show completeness we show that $\sum [B_\alpha]$ exists.
If $\mu \leq \omega_1$, let $A_\alpha = B_\alpha\setminus \cup_{\beta < \alpha} B_\beta$ be the disjointifications. Then by ccc there must be some $\gamma < \omega_1$ such that $[A_\alpha] = [\emptyset]$ for all $\alpha > \gamma$ (since the $[A_\alpha]$ form an antichain). Then simply see that $\sum_{\alpha < \mu}[B_\alpha] = \sum_{\alpha < \mu}[A_\alpha] = \sum_{\alpha < \gamma}[A_\alpha]$ and the last one exists due to ccc.
Next inductively assume we have this for all cardinals below $\mu$. And given $\{[B_\alpha]\}_{\alpha < \mu}$, by inductive hypothesis $[A_\alpha] = \sum_{\beta < \alpha}[B_\alpha]$ exists for all $\alpha < \mu$. Then I think I can just define $[C_\alpha] = [B_\alpha] - [A_\alpha]$.
Notice that here if $\alpha < \beta$ and $[A_\alpha] < [A_\beta]$ (i.e, $[A_\alpha] \neq [A_\beta]$). Then we would have that $[C_\alpha] \perp [C_\beta] $ as well. In order to keep $\{[C_\alpha] : \alpha < \mu\}$ from being an uncountable antichain then we must then have that the $[A_\alpha]$ only take on countably many different values. This means that $\sum_{\alpha < \mu}[A_\alpha]$ exists and $\sum_{\alpha < \mu} [B_\alpha] = \sum_{\alpha < \mu}[A_\alpha]$.
Does this seem correct? I was pretty cavalier about using ccc and when antichains are formed so I wanted some confirmation. Hopefully I am not making this out to be far tougher than it is.