Is there a way to vote to be able to transfer questions to other SE sites? Do moderators even have that capability?
Yes there is, and yes mods have that power. Take a look at this post: How can we migrate questions to other SE sites?
Basically, as the targets are limited, the way to go is to flag as "other" and include there the reason for flagging and the possible target you suggest. For example, a sample phrasing for this situation could be:
I'm voting to close this question as off-topic because it is not about Navigating the Workplace, and would better fit Interpersonal SE.
Given the post were not trash then one of our esteemed Mods will eventually come by and take the action they seems according to this situation.
Overall, is there a better way to handle this without forcing the OP to manually go to another SE site to ask his question?
This is a bit harder, given the target sites are limited.
Indirectly, we could prevent this situations if users took more time to read what is off-topic here, or if they previously knew IPS... as we can see, it is something hard to achieve. IMHO, it is not rare that sometimes posts placed here could better fit IPS, and thus adding it as target could help.
Although I recall there is some guideline that Beta sites should not be target for Migration Qs if possible, to allow them to organically generate their own question flow... perhaps that is the reason why we don't have IPS as target.