When does analytic in the operator norm imply analytic in the trace class norm?

Question

Consider $U$ a nice compact region in $\mathbb{C}$ with boundary $\Gamma$. Let $S_1$ b the ideal of trace class operators on a separable complex Hilbert space $H$. We will let $\|\cdot \|$ be the operator norm and $\|\cdot \|_1$ be the trace norm. Suppose $W:U\to S_1$ is complex analytic in the operator norm. Under what conditions is $W(\lambda)$ analytic in the $\| \cdot \|_1$ norm? I have proved that the following are equivalent when $W(\lambda)$ is operator analytic:

1. $W(\lambda)$ is continuous in the operator norm
2. for a fixed $M$, we have $\|W(\lambda)\|_1 <M$ for each $\lambda \in \Gamma$
3. tr $W(\lambda)B$ is analytic for each bounded operator $B$.
4. $W(\lambda)$ is analytic in the $\|\cdot\|_1$ norm.

I can provide some ideas for these proofs if that's be helpful.

This leads us to Question 1: What if we know that tr $W(\lambda)$ is analytic? Is there a nice way to compare tr $W(\lambda)$ with tr $W(\lambda)B$? I would love an inequality like $$ |\text{tr }AB| \leq\|B\||\text{tr }A| $$ for $A \in S_1$ and $B$ bounded. Although it would probably be greedy to expect this in general.

Also, I'm willing to impose even stronger assumptions on $W$ if necessary. One very strong constraint is to assume that $W$ has a rank bound along $\Gamma$, I.E. for a fixed $N$ we have rank $W(\lambda)<N$ for each $\lambda \in \Gamma$. This actually guarantees analyticity as $$\|W(\lambda)\|_1 \leq N\sup_{\lambda \in \Gamma} \|W(\lambda)\|$$

Attempting to weaken this condition, we arrive at Question 2: What happens if $W(\lambda)$ is finite rank for each $\lambda$, but has no rank bound? I suspect that finite rank and analytic actually implies rank bounded, but I do not know.

Edit 1 Here's a fun idea to that might help prove finite rank implies rank bounded. The set $$ S_n = \{\lambda : W(\lambda) \text{ has rank at most}n\} $$

is closed (continuity of $W$ tells us singular values are continuous). So Baire Category theorem tells us that some $S_n$ is dense somewhere. So in some open set, $W$ is rank bounded. So can I use an analytic extension in the trace norm to do something? This looks like the proofs of the open mapping theorem and whatnot...

Edit 2 Here's another fact that may be helpful. Consider a sequence of complex analytic functions $f_n:U\to \mathbb{C}$. Suppose they converge pointwise to an analytic function $f$. Then $f$ is analytic on an open dense neighborhood of $U$. This is potentially helpful because for any orthonormal basis $\phi_i$, $$ \text{tr} W(\lambda) B = \sum_{i=1}^\infty \langle W(\lambda)B\phi_i,\phi_i\rangle $$ And because $W(\lambda)B$ is analytic in the operator norm, so will the each inner product be analytic.

I am fairly familiar with Gohberg's work on trace class operators. Unfortunately, despite all of the great theorems on bounds for singular values, knowing that tr $W(\lambda)$ is analytic gives no information about the singular values.