I suggest that you talk to another professor.
I'm assuming you use an H-bridge motor controller to actually drive the motor. Without actually trying to implement it (that is, it may not work), I'd suggest that the controller has an enable input which will disable its internal drive. If you disable the controller just before you make the switchover, then reenable it just afterwards, the controller ought to handle the job.
Alternatively, if you provide a decent-sized capacitor at the controller input, it will provide current during the switchover. And while this is obviously impractical for prolonged outages, using MOSFET switches the switchover period should be measured in microseconds, and it's not hard to get a capacitor do that, even for fairly high currents.
Yet another possibility you might consider is to provide the two batteries at different voltages, with the high-demand battery (oddly enough) providing a slightly lower voltage than the cruise battery. Then you provide a switching circuit which looks like

simulate this circuit – Schematic created using CircuitLab
In cruise, the switch is closed, and since the cruise battery has a higher voltage than the acceleration battery, D1 is reverse-biased, and no current is drawn from the accel battery. If the switch is opened, the accel battery provides all the power.
The switch is used only to maximize efficiency. Some power is wasted during acceleration due to the drop across the diode (which is why a Schottky is specified), but this is only true during the relatively brief periods when high acceleration is called for. It also has the virtue of allowing the accel battery to act as reserve battery, just in case the cruise battery is exhausted during use.
It's perfectly possible to replace the switch with another diode. In this case, high current draw will hog down the cruise battery until the accel battery takes up the slack. This has a couple of drawbacks. The first, and most obvious, is that there is a continuous power loss due to diode heating. If you use a high voltage (and therefor lower current) battery, this will minimize the problem, but it may still be objectionable. A second, and less obvious difficulty is that as the cruise battery is discharged its voltage will lower, and the accel battery will be accessed more and more, until both batteries run out.
What you have will work, but has some issues. I don't know what kind of "SCR Controller" you have, which is a big black hole.
In the simplest terms, you could use a variac feeding your full wave bridge. That would give speed control. With that, you should have a variac that has a zero-output interlock switch. Only energize the motor at zero volts. If you energize a standard permanent magnet 90V DC motor with full voltage, it will appear as a dead short to the supply, since if it isn't rotating, there is no counter EMF to limit the current. You might be able to substitute the SCR controller in place of the variac, but with the same caveat, start at 0 Volts.
The best solution, which should be pretty inexpensive for an under 2HP motor, is to just purchase a DC motor controller. A quick look shows a bunch available on Amazon for under $40. A DC Motor controller will generally have an inherent acceleration/deceleration ramp setting, current limiting, and possibly even a setting called "IR Comp", which when properly set will allow the motor to run at a virtually constant speed independent of the load. EDIT TO ADD: The simplest DC Motor controllers in the 90VDC class are basic phase controlled SCR's (with additional intelligence for current limit, ramps, etc). Generally, these are also the most reliable. There are other PWM controllers for 90/180VDC motors which also work well, and they are also surprisingly inexpensive.
Best Answer
Use the MOSFET. Its use case is switching a DC load. If the LED strip has a lot of inductance, put a flyback diode across it.
If you drive the gate with 5 V relative to source, then the RDS(on) is specced as 0.077 ohms from the linked data sheet. The power dissipated will thus be 0.27 * 0.27 * 0.077 = 0.006 W. No need for a heatsink.