I can see where you're coming from, but my point is that we'd build an amusement park, and then... what?
We'll be stuck to a reactionary role if we don't want to start suspicion about us. I suppose there's building bigger parks and expanding outwards, but sooner or later we're going to be attacked, either by gangs or even by Endbringers, and then there's Coil acting around in shadows.
I don't mind us building an amusement park, but I feel it's something to use to increase our PR once we have our basics down. A talk with the Mayor would take time to get anything done, let alone the actual time needed to announce the building, advertise, and actually build the thing.
Additionally, there'd be the risk of the PRT coming over with some claim about how Tinkers aren't allowed to build without authorization, and how we'll have to register to ensure public safety or whatever, and so on.
Finally, we don't really have a story they don't want us to tell, do we? It looks to me like the other way around, where them claiming we brought down the barrier could endanger our position with the mayor. I doubt us saying it was for the sake of a robot would win us any points.
Meanwhile, if we take down Coil and announce it was for the sake of the city, what is the PRT going to say? We did it to capture a lair? What do you mean, everyone knows Coil doesn't have such a thing, he's just small-time! Any claim about us bringing down their shield would be suspect, especially in light of us doing a 'public service' by taking down a gang.
Offering the Mayor an amusement park right afterwards would work much better, especially since we'd have given him a reason to trust us. The lair we'd have by then can be kept a secret as a fallback option, or to work on alternate operations in.
We'd have more choices, a higher chance for success, and materials for future use, all while everyone thinks we're just an amusement park manager that can't even fight a real gang.
Edit: That got longer than expected.