Clearly, your business partner feels very strongly that your approach has some deep flaws and just won't work. If they didn't, they wouldn't be suggesting starting from scratch or abandoning the project entirely, especially not at this late stage.
What you should try to do first is find out why they think your approach won't work. This might be tricky, since their opinion might be based on just a "gut feeling" (to be fair, so is your opinion that it will work) which they may not be able to immediately explain in a clear and convincing way. They might also feel reluctant to get into a technical argument with you on a topic where you have much more professional experience, and where they may thus feel they're at a fundamental disadvantage just because they don't know the right jargon and buzzwords and can't lean on their authority from years of experience like you can.
So your first task will be to convince your business partner that you don't want to argue with them or to convince them that your approach is better, but simply to understand why they feel it's problematic.
You might want to start by telling your business partner that, if they think there's a problem with they way you've designed the territory acquisition mechanic, then you believe them. But also say that you can't really fix the problem unless you understand it, and that you'd like to try to help them analyze and explain the issue so that you can then find a good solution to it together.
If they bring up their previous suggestion of using fixed 5km tiles, say that you feel they have their own problems (as you've mentioned in your question above) and that, while it might be a possible backup plan, you'd really like to come up with a solution that still preserves some of the advantages of your original multi-scale solution, if at all possible.
In this case, be prepared to also have the same conversation from the other side, i.e. to try and explain to your business partner why your gut instinct is telling you not to go with the fixed-size tiles and what specific aspects of your original approach you feel would make the game more interesting and playable. And try to do that non-confrontationally, in a way that your business partner can understand and relate to, without unnecessary jargon and without leaning on the "I'm the expert so I know what works" argument. Remember that you're not trying to convince them that they're wrong; you're trying to make them understand where your gut feeling is coming from.
Anyway, if you manage to have that conversation and explain the sources of your respective gut feelings to each other, what you'll probably find out is that each approach has issues that one of you considers to be likely showstoppers. It's even possible that both of you are right, and that both approaches do have major problems.
However, with a bit of luck, you may also be able to identify a list of features that you both do consider desirable, and possibly even come up with a third approach that keeps as many of those desirable features while avoiding the showstoppers.
I won't try to suggest any such third approach here, both because I don't know enough about your game and also because it wouldn't really fit into the scope of this answer anyway, but I do wish to point out a few implicit assumptions that you both seem to be making, and some possible ways of breaking them:
You're both assuming that the tiles that players can claim should be regular squares. What if they weren't? What if the tile shapes were instead irregular, and maybe (more or less) aligned with real-world map features? What if the tiles could have different sizes (even at the same "zoom level")? What if players could somehow move tile boundaries, or even join or split tiles?
You're both assuming that players can or should only interact with other players at the same "zoom level". What if players could have hierarchical relationships, with smaller-scale players being able to act as (more or less temporary) "vassals" of higher-scale players? Or what if players had to team up to claim larger tiles?
You're also both assuming that tiles should be claimed one at a time? What if placing a unit e.g. claimed all tiles within a certain radius of the unit (with some sort of nearest-neighbor and/or first-come-first-served rule to handle cases where the same tile is within range of multiple units)?
Ideally, with enough thinking outside the box, you'll be able to work out some new solution that isn't just a compromise between your opposing viewpoints, but actually something that you both consider better than either of your original approaches.
Of course, it's also possible that this won't happen, and that your viewpoints turn out to be simply irreconcilable — or that, even if you do come to understand each other's objections, you simply can't find any solution that would address all of them.
In that case, you may simply have to pick one option and go with it, flaws and all. Or, of course, you could also pick neither and just abandon the game — but so late in the process that seems like a bad idea. Sure, you don't want to fall into the sunk cost fallacy and continue with a doomed project just because of how much you've already invested in it, but you also don't want to abandon a nearly finished game just because it might not be as good as you hoped. In the end, at least in purely financial terms, releasing a crap game is still better than releasing no game at all.
One option might be for one of you to agree to assume full control of the project and of all costs and risks going forward, with a correspondingly higher share of any future profits. (Obviously, any prior investment should still entitle the other party to a share of the profits as well, unless they're bought out fully.) Depending on the agreement you come to, that might involve one of you starting to pay the other for any remaining work, or of one of you quitting the project entirely and the other one having to bring in additional people to finish any remaining tasks they can't complete on their own.
In general, that's also the point where you probably ought to involve a lawyer or two, if you haven't already. Still, if done right, this should allow one of you to continue with the project as they envision it, with the other one retaining a share commensurate with their past investment but no obligation to sink any more time or money into the project. If you really cannot reconcile your visions, that's really the best you can hope for.
(OK, one more option would be for you to find a third party willing you buy your whole game project in its current form and to assume control of it. Depending on just how promising and close to being finished your game is, that might be possible.)
