When will you buy the BV/CR instead of Minion or BV/Minion?
You need to buy at least 6 Minions to make this work. Then you need a Remake and another buy on BV/CR. These are at least 7 buys with more than $4 each. I might be wrong, but you do need more than 10 turns to make this happen here. And with only 6 Minions, you cannot have more than $10 from your Minions because at least one Minion must be used to discard. So I think, you would start to buy Provinces by turn 12 if all works out well - resulting in at least two double Province turns.
But then, the two Tunnels are worth 4VP, the 3 Estates are worth 3VP.
The BM player could start buying Points (not all Provinces, though) at about turn 7. He will get more Golds by discarding Tunnels. He might even buy Tunnel over Duchy for more Golds. Reaching 4 Provinces before turn 13 seems to be possible, even if you have only 4-card hands from turn 10 on. Among these 4 cards might be two golds and a terminal. It doesn't have to be each turn. Just every other turn would suffice. I'm really tempted to doing this.
Anyways, the Familiar would totally kill this Strategy.
Tunnel isn't great against Minion, since almost half the time you'll draw them in your 4-card hand and that sucks.
It doesn't suck if your three other cards are 2 Golds and a terminal.
You shouldn't buy too many Tunnels, though. One or two are just fine as far as I have tried these kind of "counters".
Also, in a Minion deck, actually getting rid of stop cards is better than replacing stop cards with slightly less bad stop cards. I'm not sure if Remake or Spice Merchant is better for that; Remake lets you trash two Coppers per play, but on the other hand, Spice Merchant trashes one and still lets you buy something that turn.
In the beginning, a Spice Merchant who thins down a Copper to gain 2 cards nets you only $0.6 = 2*$0.8 - $1. If you trash it for the $2 you get $1. Either way I wouldn't expect you to hit $5 unless you are lucky or you have bought 3 Silvers. So I'm pretty sure you shouldn't be able to buy Minions consistantly before turn 5, maybe even later. Like in most setups, I would prefer trashing Estates first, and gaining Silvers for Estates is really good for your economy. Even on a "not so lucky draw" with 1 Estate and 3 Coppers you can thin down one Copper and exchange an Estate for a Silver.
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edit: I've tested this against the Bots. I was able to win all games except two. One game I lost due to rotten draw luck and misplaying it on top. The other game was a draw.
You don't get as many Provinces as I expected. nevertheless you can deny Border Villages and thus 3-pile before the Minion player is ready. It
is risky, because you have to get at least one gold early on to be safe. On the other hand I won a game where I got the first Gold on turn 9. If your luck is bad, you can at least pile the Tunnels, sneak a Duchy or a Province every now and then and end the game quickly on a 3-pile. If you're doing well, you'll get so many points, your opponent cannot catch up any more.
This strategy worked out even if the Bot baught a Familiar. Yet, it was more diffcult, if he did.
I think opening Navigator/Tunnel would not be smart as this would signal what you're going to do. Opening Remake/Oracle on the other hand might work. Remake your Estates into Tunnels and catch the Navigator later on. If your opponent goes for Familiar, you can do some more enginy stuff where Remake was enough to ignore Familiar as we have seen.
Border Village actually helped me more than the Bot as I can use it to 3-pile while getting Duchies. Most games I won with only 1 or 2 Provinces. If your opponent goes for the Minions, you can safely load up on Tunnels as they provide three benefits: they can be discarded for Golds, they provide points and they are a pile. You will have enough Golds to pile the Duchies, maybe even the Border Villages and the Duchies.