Thompson, 1987, doesn't address this directly, but did consider an algorithm for specific configurations of the probabilities. He was looking for the worst case scenario when the bounds of error were not constant, and suggested a search over many configurations.
Here is my modification of Thompson's algorithm. This version reduces the search to your a priori configuration and, perhaps, some near neighbors.
Select a set of vectors $P =(P_1,P_2,P_3,P_4,P_5), \sum_j P_j=1)$, which are plausible according to your a priori beliefs. You state that you have one configuration in mind, and it is likely that small departures from that configuration will have little effect on the calculation. Still, I recommend that, for safety, you consider plausible departures from this configuration.
Choose an overall $\alpha$ level and bound on error $d_j$ for estimating $P_j$. Your question has $d_j = 0.05$ and $\alpha = 0.05$.
If $p_j$ is the sample proportion for category $j$, a simultaneous $1-\alpha$ confidence for the $P_J$ will be of the form:
$$
\left[\,p_j -d_j,\, p_j +d_j\,\right]
$$
with the desired property that
$$
\text{Pr}(|\,P_j - p_j\,| > d_j\text{, for any j}) \le \alpha.
$$
Compute the worst case $n$ for your $\alpha$ from the table in Sample size for categorical data
Starting from your initial configuration of probabilities , compute $\sum \alpha_j$, where $\alpha_j = 2(1-\Phi(z_j))$ and
$$
z_j = \frac{ d_j\sqrt{n}}{\sqrt{P_j(1-P_j)}}
$$
($ 1- \Phi(z)$ is the probability that a standard Normal variable is $\ge z$.)
If $\sum \alpha_j< \alpha$, repeat with a smaller value of $n$. If $\sum \alpha_j>\alpha$, repeat with a larger value, until the smallest $n$ is found such that $\sum \alpha_j \leq \alpha$
- If you believe in your initial configuration, you can stop there. Otherwise repeat for every vector of probabilities in your plausible set and choose the largest $n$.
Reference
Thompson, Steven K. 1987. Sample size for estimating multinomial proportions. The American Statistician 41, no. 1: 42-46.
