Go For It
I wouldn't have any problems motorizing that bike. If there is a weak point, my guess would be the wheels, not the frame. Since these kits are all set-up to drive the rear wheel using the spokes, you just want to be sure that the rear wheel is properly tensioned. I'd also put the largest size tire you can find that will fit in the frame and fork on it as well. If the wheels are 700c you will have a lot of options for larger tires, but if they are 27in., your choices are limited.
If what you really want is a beach cruiser you could sell your current bike on your local craigslist. Advertise it as a good beginning for what is called a fixed gear bike/project. Older lugged road frame sets with horizontal dropouts are selling for a preimum these days due to the fixed gear craze. Or offer it for trade for a good quality beach cruiser.
One other thing if the seat height as shown in the photo is your correct seat height, the the bike is way to large for you, and you might want try and trade it for somethig that fits better.