I was going to post after the game, but I don't really see it as a spoiler now.
The way to have handled Suttree requires more depth than noting which units are effective counters to him, it's all about the strategy of containment. So city placements, research aims, every needs to be designed around the minimum necessary to contain Suttree regardless of what he does to hurt you. Then you are relying on his metagame skill to understand that harming you gains him nothing, but he has other options that are "kind of" mutually beneficial for the both of you, namely other directions to expand. Unfortunately, this is where I can't really comment because that would contain spoiler knowledge, so I'd have to come back to this once the game has ended or you guys are dead.
What I would do in your theoretical scenario is note that Suttree has invested 750 hammers into cats, 630 hammers into axes and 245 hammers into spears, for a total of 1625 hammers, and Construction as a tech. Personally, I'd have either rushed to HBR or Construction (I think you messed around with Alphabet?) most likely HBR as my first Classical era tech if I knew I needed to spend 1000's of hammers on units, because those hammers would be able to cover a significantly larger area and to react to others as needed. That would be realised, as you said, by potentially greater gains against mh (and long term weaker mackoti). A combination of chariots, HA and bowmen will deal with any and all barbs (I do agree that bowmen were a good choice regardless, because they will always deal well with barbs). The chariots and HA can be trained to 5XP and rotated out for new ones, this limits the cost of city improvements, but the main unit producers should still have gotten cheap stables.
Ideally, I wouldn't have built more than 2 axes, but that's just me (even in PB15 as AGG I think I built 4 axes and 4 swords in the entire game). You pointed out that a few HA aren't enough, and I agree, you do have to just go overboard on building them. The other advantage to that though is that if Suttree moves in his 1600 hammer stack, and you are sitting there with 1200 hammers of HA (that's 24 HA, similar in size to the stack I had in PB15) and 500 hammers of cats, either his stack dies, and you get a GG or two, or you just run past him and take all his cities. Most likely do both. You only need to remove the cats from the equation and he can't get odds on you cities, and F2 HA have 50% chance to withdraw. So you could either go for the kill or just try to flank away the cats, and he can't get odds on bowmen in cities; in fact after a few flanked cats, you might get reasonable odds with a cat or two and could then throw bowman at him with odds.
I also think these numbers highlight another cognitive dissonance; you didn't actually try to contain Suttree. If you're saying that he spent more than double the hammers you did, you were only trying to manage his threat, but you were leaving him the continued option to build up and try to invade later: you were reacting to his threat, rather than being proactive and forcing him to defend. The downside is that being proactive in axe v axe combat is basically down to stack size and presence of cats, and he had nothing better to do than build axes.
Really though, I don't think this is a great answer, but I can't really remember your early game decisions, and I think that a lot of the "failing" against Suttree comes because of trying to deal with the capital you had and the lack of early workers.
The way to have handled Suttree requires more depth than noting which units are effective counters to him, it's all about the strategy of containment. So city placements, research aims, every needs to be designed around the minimum necessary to contain Suttree regardless of what he does to hurt you. Then you are relying on his metagame skill to understand that harming you gains him nothing, but he has other options that are "kind of" mutually beneficial for the both of you, namely other directions to expand. Unfortunately, this is where I can't really comment because that would contain spoiler knowledge, so I'd have to come back to this once the game has ended or you guys are dead.
What I would do in your theoretical scenario is note that Suttree has invested 750 hammers into cats, 630 hammers into axes and 245 hammers into spears, for a total of 1625 hammers, and Construction as a tech. Personally, I'd have either rushed to HBR or Construction (I think you messed around with Alphabet?) most likely HBR as my first Classical era tech if I knew I needed to spend 1000's of hammers on units, because those hammers would be able to cover a significantly larger area and to react to others as needed. That would be realised, as you said, by potentially greater gains against mh (and long term weaker mackoti). A combination of chariots, HA and bowmen will deal with any and all barbs (I do agree that bowmen were a good choice regardless, because they will always deal well with barbs). The chariots and HA can be trained to 5XP and rotated out for new ones, this limits the cost of city improvements, but the main unit producers should still have gotten cheap stables.
Ideally, I wouldn't have built more than 2 axes, but that's just me (even in PB15 as AGG I think I built 4 axes and 4 swords in the entire game). You pointed out that a few HA aren't enough, and I agree, you do have to just go overboard on building them. The other advantage to that though is that if Suttree moves in his 1600 hammer stack, and you are sitting there with 1200 hammers of HA (that's 24 HA, similar in size to the stack I had in PB15) and 500 hammers of cats, either his stack dies, and you get a GG or two, or you just run past him and take all his cities. Most likely do both. You only need to remove the cats from the equation and he can't get odds on you cities, and F2 HA have 50% chance to withdraw. So you could either go for the kill or just try to flank away the cats, and he can't get odds on bowmen in cities; in fact after a few flanked cats, you might get reasonable odds with a cat or two and could then throw bowman at him with odds.
I also think these numbers highlight another cognitive dissonance; you didn't actually try to contain Suttree. If you're saying that he spent more than double the hammers you did, you were only trying to manage his threat, but you were leaving him the continued option to build up and try to invade later: you were reacting to his threat, rather than being proactive and forcing him to defend. The downside is that being proactive in axe v axe combat is basically down to stack size and presence of cats, and he had nothing better to do than build axes.
Really though, I don't think this is a great answer, but I can't really remember your early game decisions, and I think that a lot of the "failing" against Suttree comes because of trying to deal with the capital you had and the lack of early workers.