Need some help picking a relay
The timer has three contacts- two of which are high current and act as a permissive chain such that the blower motor is powered preferentially.

You could replace the timer with an electronic device  controlling a contactor  with at least three poles, rated for the maximum current the heater and motor draws together, plus some safety margin- maybe 35 or 40A. These contactors are widely available, but they do tend to create a lot of electrical noise when they switch (which can screw up your micro, but is outside of the scope of this answer). 

Personally, I would not do this, given the risk-return ratio. I would certainly not use a solid-state relay, given their propensity to fail 'on'- this thing only has to ever operate a few thousand times in its life and any contactor  can do that reliably. The one you linked to is not really an SSR- it's something else and is unsuitable. 

I also note that your schematic shows the neutral connected to the chassis. I believe this is an old (and now considered unsafe) connection method that was grandfathered for a period of time. If the NEUTRAL becomes disconnected the USER can be electrocuted, generally considered to be a bad thing.