Welcome to the site and this niche of the hobby.
You are fortunate in your location in that there are many retailers that will allow you to see and sit on their bikes. As you may know, fit is important especially if one is not a standard size.
You are unfortunate for your budget. Complete e-bikes start about $1500. Due to the occasional lemons out there that stop mysteriously stranding one, and the inability to test used batteries, used e-bikes are not a bargain. Lithium Ion battery life is 5-8 years in the best cases.
In October I paid $630 for a battery that worked from Luna. Two $300 batteries from Amazon & e-bay didn't work. I got one refund, from Amazon. You only have 31 days to decide something is wrong. You can't ship batteries back, it is illegal. My battery is 17.5 AH 48v and you may not need that much. The average battery that comes with a bike is about 10 AH. Cold weather range is about 1/2 of warm weather. I got about 40 miles out of my battery on a 40 deg day.
In this late day I would not put up with a bike that has rim brakes, since they are fairly useless in the rain. Disk brakes are quite competent in rain. I have the cheaper mechanical disks with small 160 mm rotors which I find have only one disadvantage. With 2500 miles on them, adjusting them tighter to make the thinner pads work right means the force of the spring is higher. However, I don't have to bleed them as I've done hydraulic brakes on cars. My brakes were fine at gross weight 290 lb down a long 10% grade last month.
I use my bike for cardio exercise, using the electricity only if the wind is over 12 mph in my face or the ride is > 25 miles. You can't get much exercise with most mid-drive bikes, your energy spins the motor in PAS level 0. Exceptions are Yamaha and one model of shimano mid drive. You can pedal those. DD hubs are on the cheapest bikes, mine left drags me down about 25% power off over no motor at all. Geared hub drives don't drag at all, they have a one way clutch. Court's reviews on the main site will tell you what motor, brakes, and many other features, are on a bike. In NYC, e-bikes with throttles are still illegal, so watch to avoid that feature. NYC confiscates illegal e-bikes.
Among the cheapest bikes I think are Magnum & Juiced. They have cheap hub drives. 10000 miles may be wear limit on a geared hub drive, but then they only cost $250-500 for a power wheel and most of them for rear fit standard 130 mm brackets.
Happy shopping.
