I also just realised that any idea you have since your Dominion: Adventures play-testing means that there is no similar card in the new official set. BAM!
Fact. Now all I have to do is propose every possible card that's not in Adventures and you'll know exactly what's in it!
Let's talk about the new Bookkeeper first. This looks promising. I even wonder why such a card isn't there already. It will always be a little nasty and sometimes really screw over your opponent. Which makes me think it should cost $4 but that's just my intuition. I feel like it's often at least as good as Cutpurse in the early game but stays strong later. Even early, if your opponent decides to put back an Estate rather than discard a Copper, it surely screws up their next hand.
I think the fact that they often get to choose early whether to topdeck an Estate or discard a Copper makes it weaker. It's sort of like a reverse Coin token. If they don't want to be bumped down $1, their next hand suffers for it. But they can choose whichever is more advantageous. But I might make it cost $4 just because the set is hemorrhaging $4 cards and has way too may $3 cards. For $3 right now, it has (new) Bookkeeper, Convoy, Floodgate, Gambler, Mill Town, Refurbish, and Tinker. Assuming I remove the old Committee, it has Craftsman, Dignitary, Profiteer, and Terrace at $4. Bookkeeper and Tinker would probably be fine at $4, although I kind of like Tinker at $3. Refurbish has always been a prime candidate for cutting just because it's 100% off-theme. As much as it saddens me, Mill Town could also be cut. It's a fine card, but the set has a lot of villages (Jubilee, Mill Town, Terrace, General), and Mill Town is the one I need the least.
Conquest is a potential Axeman replacement. I may try a cheaper ($4?) Barrister that makes players trash Domains from their hand.
Conquest: I am a little surprised you're going for a harsh trashing attack. But I guess since it's a one-shot it's alright. Gaining and playing the gained card immediately afterwards is a good idea. I agree with Awaclus, though that hitting only Silvers with an expensive one-shot would really suck. And there are plenty of Silvers in an average Dominion deck.
I've got an extraordinary suggestion: try Conquest targeting cards from $4 to $7. It would look quite unusual but it changes a lot and would make me rather think "this is gonna be a successful Conquest!"
I also like the version that immediately plays the gained Action card, and twice if it costs less than Conquest, but Pacovf made a valid point in that nobody wants to be forced to play a trasher. But you could make both plays optional! I guess that's too many words for your liking, though.
I'll keep the $4 to $7 option in mind. I do worry that it's easier to trash Provinces with, though. One Highway is all you'd need.