> there could never be an area of an infinitely large map that had five regions each needing a different colour
A more precise version (infinite -> finite, "...needing a different color looking only at that sub-map) is plausibly true. That doesn't help you though, because...
> maybe the 'outside' colours are the same - but in that case you could always just recolour them individually before putting them together
...this is false. Five patches of one sub-map might touch one patch of the other sub-map. So now you'd have to prove that the first sub-map can be colored with three colors to rescue your proof of the FCT. Further...
> you take two such sub-maps, put them next to each other [...]
...your argument here seems to boil down to "I can construct the full map by gluing together sub-maps". Depending on how you constrain the sub-maps to prove necessary properties (see above), this may not be true. But even if it were, your argument that it works seems to only consider gluing together two sub-maps of specifically size five, which obviously does not allow you to construct maps with more than ten patches.
This makes for two to three fully deal-breaking errors in your proof sketch.
> there could never be an area of an infinitely large map that had five regions each needing a different colour
A more precise version (infinite -> finite, "...needing a different color looking only at that sub-map) is plausibly true. That doesn't help you though, because...
> maybe the 'outside' colours are the same - but in that case you could always just recolour them individually before putting them together
...this is false. Five patches of one sub-map might touch one patch of the other sub-map. So now you'd have to prove that the first sub-map can be colored with three colors to rescue your proof of the FCT. Further...
> you take two such sub-maps, put them next to each other [...]
...your argument here seems to boil down to "I can construct the full map by gluing together sub-maps". Depending on how you constrain the sub-maps to prove necessary properties (see above), this may not be true. But even if it were, your argument that it works seems to only consider gluing together two sub-maps of specifically size five, which obviously does not allow you to construct maps with more than ten patches.
This makes for two to three fully deal-breaking errors in your proof sketch.