I think you guys are in a little bit more dire of a situation then you've acknowledged.
- You have no metal / horses
- AI has both metal and horses already in play
- The only way your archers will get decent odds is being attacked, but the AI will only attack you if there's a very significant chance of taking your city, or he has good odds against your unit on a random tile.
- So you can defend, and then he will go pillage mode striking deeper into your empire. You don't have enough roads or the geography to really outplay him, so you'll end up having to just barricade every city with a bunch of archers, and he will pillage you out of the game.
- To top it off you have to get a settler to the two fish spot asap, which I assume would come from your capital and cost you a couple potential defensive archers.
I'm not sure how you plan on getting him to attack you on the gold mine to suicide his stack. The gold mine is on a hill and you do have archers, and he has only one axe, so if you defend it he will probably just walk around. And in general trying to coerce him to attack you here is weak, because you only have archers to clean up the mess, and he would then be defending from the hill tile.
Similarly I'm not sure you can get him to suicide your city without taking a big risk at losing the city.
I've been in this situation early before with k-mod and the way I usually handle it is abuse the way karadoc intended to make the AI stronger. It will no longer suicide units but enter pillage mode looking for vulnerable cities. This is actually a huge vulnerability. By switching whether a city is vulnerable / strong you can often get the AI to walk back and forth forever, or at least until reinforcements change the dynamic.
Stalling can help you get more units/techs. Sometimes my goal is to not fight a single battle and just stall until I can bribe them for peace.
Feel free to not go with that approach, as it's unfamiliar, very risky, and very gamey. But I'm not sure how you get out of this pain free, without some good luck or stalling of some kind. One thing is for sure, I'm glad I'm not the guy who has to do the next set