I would have to see the website. It could also be a case of just needing a Gen Z makeover. This is a common story, did a site this week for a person in this exact position. You have a few great things going for you, eagerness and budget. I would start by hiring a plumbing sector PPC guru. Get your FB leads maximized with the first budget spend if you haven't, this has quicker effect. I would try to find someone who specializes in Plumbing PPC and does just that, all day, every day.
Then make sure you have strong content, and local content velocity. Hit many points rapidly, the current "rank in one" day method involves creating a grid of articles and deploying across 10 platforms all at once. This practice might be frowned on here, or considered "parasite", I'm not sure — but it's working right now. If you need to get seen fast, it's a way that works without having to buy links, etc.
Here's the twist. If your competition is buying links, then what? I'm currently soul searching on this exact point so I have no answers but it comes down to competing to win. If you want to win? Only option is to do things that creates wins. That boils down to dunking on whatever your opponent is willing to do to beat you — which then becomes the ante for beating them.
It's a like a hockey game where one team is playing dirty, the other team has to play dirtier. It's the only solution we have if we need the win. Fundamental human nature. Godspeed.