For a cover to use on the Bulldog Powered Drive Trailer Jack # BD500199 referenced in your question, I recommend the Ultra-Fab Protective Cover for Electric A-Frame Jacks # UF38-944020. This will easily slip over the jack and can be secured using the drawstring. This cover measures 12 inches x 14-1/2 inches. I have attached a video demonstration of the cover for you to check out.
It sounds like the Bulldog A-Frame Jack, part # BD500200 is just what you are looking for. The BD500200 has an outer tube diameter of 2-1/4 inches which matches what you are replacing and this jack has 2 toggle switches. One is used to control the up and down movement of the jack and the other switch turns the LED lights on and off. The mounting plate features mounting holes that allow you to mount it so the controls are at the front of the jack or mount it so that the controls are...

The efficient ball screw on this powered jack ensures smooth, quiet operation. Easy-to-access toggle switch and bright, side-mounted LEDs make setup a cinch. Durable, corrosion-resistant jack offers Features: Electric jack with 12V motor lets you lift and lower your A-frame trailer quickly and easily Efficient ball screw makes it fast and easy to position your trailer Ball bearings roll along
The Camco Electric Tongue Jack Cover helps protect your electric tongue jack when it is not in use and in storage. The durable 3-layer spunbond polypropylene provides protection from harmful UV rays, excess moisture and dirt and debris. Tongue Jack Cover accommodates most styles and brands of Electric Tongue Jacks, and measures 18" tall x 7.5" wide x 8.5" deep.
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