With most controllers you can set a rate of rise in the settings, so you can have a controlled rise to temperature over a period of time. Then use the alarm output, which is also programmable on them, to start the separate timer. At the end the timer simply turns off the temperature controller, allowing the cooling to ambient to occur.
Available on BTC and Omron PID controllers, and you just have to get the version with alarm output to use it, or open the controller on those with that as option and add the small 24VDC low current relay on the board.
For sanity sake I would also recommend getting a datalogger and using it to measure the profile for every run, so that you will pick up almost immediately if there is a cycle where you did not reach setpoint, or held it for too short a time. As I assume you will be doing this cure overnight there otherwise would be no indication that there was a fault otherwise. 2 channel logger to measure both temperature and the power to the heaters, so you can see duty cycle on them, and this will tell you if one fails from the increased duty cycle with otherwise no other indication.
As to the programming interface, what else do you expect with only 3 keys to set everything, simply make a copy of the manual and stuff it in the control box, so that it is easy to refer to it when needed. There are controllers that you can program over RS 445, but that leads to another entire level of suckiness on the horrid clunky controller programming, which often required WinXP ( at a defined service pack level as well, nothing newer than SP1 in some cases) and a special version of IE with customised ActiveX controls for it, and a real serial ( no FTDI or Prolific, has to be genuine 16550 UART register set on the south bridge) port to actually work.