Some have said the 3 speed Sturmey isn't durable under engine power. I have no personal experience with it.
IMHO, avoid internal gear hubs (Sturmey, Sachs, Nexus, Sram) and use a derailleur with your motor and jackshaft. It's must cheaper to replace the worn or broken parts on a derailleur setup.
If you do not have a derailleur hangar on your frame, you can go with an axle mounted gear changer.
I do dig the internal geared hubs (I have a Nexus on my pedal powered beach cruiser). I just question their durability and repairability when used under engine power.
I've never ran a 3-speed hub under power. The hub mentioned has gearing which range from 22t to 13t. The eight-speed cassette ranges from 34t to 14t. The 22t in the cassette's cluster is about third gear.
Therefore, using a 3-speed hub is like starting out in 3rd gear with an 8-speed cassette. However, gearing on an SBP shift kit can be as low as 34.49 at the chainring. First gear with a 34t cassette is a stump-pulling 32.57:1
A China engine with this gear would leap off the line at a standing start.
With your 3-speed hub and SBP gearing, first gear with 36t chainring and 1.33:1 = 28.03:1, second gear = 21.07:1, third gear = 15.81:1.
With a 30t chainring, first gear = 33.63:1, second gear = 25.28:1, third gear = 18.96:1.
You will have excellent power from a stop with 33.63:1 and okay power from a stop with 28.03:1.
My new cassette and derailleur cost less than $80. It is not a heavy-duty, bulletproof, top-of-the-line model. I have broken many chains until I solved that issue.
However, the Shimano cassette and derailleur have endured through powershifts and abuse for over a year. I inspect them regularly, and they show no signs of abnormal wear.
I am simply amazed by how OEM drivetrain components live reliably under the abuse of motorized power.