In my opinion you are overthinking it. Since you are a plumber and you offer your service on the site, you should go for keyword plumber. If you had website about plumbing, for example with some guides etc., It would make sense to go for a plumbing.
Anyway do not focus strictly on keywords, focus on your target group and their search intent. Imagine what they want to see in SERP.
This is old SEO thinking. You should be optimising your site for topics, not just select keywords.
Not to mention a page can rank for more than 1 keyword, so you could technically rank for both in the same page.
Plumbers fall under the category of plumbing anyway – you need a plumber to provide plumbing services.
In order to improve your chances of ranking well you need to signal to Google you’re an authority in that niche through your websites content. You can’t do that by only trying to write about a small set of keyword(s).
Don’t be afraid to write about as much as possible to do with the topic that a customer would want to know about, because it helps signal to Google you know your shit.