Should I 301 redirect that one page on Website A to the domain of Website B? Will this help Website B rank for that term?
it should, if you do everything correctly technically, help website B rank for that term.
But will it ever rank as high as Website A ranks for that term? You are rolling the dice.
How many backlinks does that page on Site A have that are related to that service? Do you think you might be able to get those people who link to Site A (and have links relevant to that subject) to change their links so they point to site B?
Here is my experience with something very similar:
My page on Site A was seeing declining rankings over the year, probably because it didn't relate to the rest of the content on the site (similar to yours), and probably because i didn't do ANY link building.
I had a similar page on Site B, but it wasn't ranking well (similar to yours).
I finally in mid-February removed that page and 301 redirected to that page on site B.
Within a few months the page on Site B was getting something like 30% MORE traffic as it did on Site A...
But only around 35% of the traffic for that page LANDS on it from search engines. The overwhelming majority of traffic lands on another page and then visits the page. Despite that fact, it IS the most popular page on my site.
So I don't know what to say, but that you should be prepared to do some heavy duty link building to supplement the 301 redirect.
I hope this helps.