Stephen: You have a lot of initiative to attempt an engine switch such as you describe. I've had my issues getting "foreign" engines wired correctly to a Cub Cadet harness. A few questions and a comment or two.
1. Does the starter stay engaged the whole time the engine is "running?" (I'm thinking it probably does, which is not good for the starter.)
2. What ignition key are you using? (Is it the original 1864 key?)
3. Do you have access to wiring diagrams for both the 1864 and the donor tractor, the Exmark mower? (The 1864's diagram is in the Wiring Diagrams available from the Manuals section on the Main page of this forum.)
Comment: There are basically two ignition schemes found on garden tractors, a positively powered ignition system such as found in an automobile in which the ignition is fed 12 volts with the key in either the "Run" or the "Start" position; and the Magneto-style ignition in which the key only engages the ignition directly in the "Off" position by connecting the "kill" wire to ground. In both systems, the charging current from the alternator (or generator) is fed back to the battery via the key switch.
Another factor that may be affecting your situation is that often the interlocks for the seat switch, reverse gear, etc. are positively powered for a positive system, that is, the switch makes or breaks the +12 volts to the coil. On a negative system, the interlocks make or break the ground to the "kill" wire. The PTO is almost always positive, especially for a Cub Cadet which typically uses a PTO that is designed to employ the block itself as a ground. (Hence the reason for all the relays; the relay allows a positive voltage to be turned into a ground signal.)
The 1864 came originally with a Mag 18 that used a Magneto-style ignition system. Its key should have communicated a ground to the magneto's primary coil in the the "Off" position. Your tractor actually has two (2) seat switches, a reverse switch, and a Reverse Relay in addition to the usual Brake Switch and PTO switch. Not all the interlocks are involved with the ignition circuit, but for a diagnostic, I would try to figure out what wires are needed to:
1. Start
2. Run
3. Stop
by-passing all other circuits, just so you can figure out what is going on; then add the other circuits back in one-at-a-time.
I realize this is a long answer to a simple question, and probably contains information a person such as yourself is probably already aware. But if I were in your situation I would try to simply the circuit as much as I could until you can figure out what is going on.