For hill climbing without the ability to change the gear the motor is in you're going to have to pick a lower gear than it was designed with, somehow, and accept that it won't help much on the flat.
The "chain driven" kit you're looking at will only work while you're pedalling, as it requires you to keep the chain tight against the motor sprocket. Despite the bike being shown with gears, the motor will only work in one gear, otherwise the chain won't hit the sprocket on the motor. If you're lucky you will be able to choose the gear, and you can choose a fairly low one to go up hills in. If you're even more lucky the other gears will still be usable without the motor.I suggest you ask the seller those questions before buying. They say:
This kit is designed for normal bike, not for the bike that can shift gears. However, you can install the kit on the left hand side if your bike is 18speed/24speed
Note that that kit does not come with a battery or charger.
The hub motor kits normally are easier to install, especially if they come built in to a wheel already (I suggest you do this, the extra shipping is usually cheaper than the cost of getting a wheel built). Since the hub motor is already set up to drive the wheel, you don't have to adjust anything that's got a lot of force or power going through it, so it's less critical that you get it exactly right. With a hub motor you only have to get the anti-rotation setup fixed properly. The usual problems are getting the wiring to work - cheap kits will almost always work, for a while, in the workshop. Getting them to work reliably in all weather while riding the bike can be a challenge. It will help if you're willing to solder connectors in place, and you should definitely plan on buying an extra hundred cable ties to hold everything onto the bike.
I think your budget is optimistic. Normally when you find a product that's much cheaper than the competition it's because it's junk. Either it's going to fail much sooner, it doesn't work as well, or it's missing key things you assume are required. In this case I think you'll find that it's both unreliable and less than you expect - the motor will be less powerful and the battery smaller (it will go less distance per charge). It will all fall apart within a few months.
Make sure that you have a reasonable workshop and a collection of bike brackets and stuff that you can use to assemble and install the kit. Those kits either come with a huge bag of brackets and bits that you won't use for your bike because they come with every bracket you might need for any bike, or they come with one set of parts and inevitably you need a different set. Cheap kits will always be the latter type, so you will need to put their brackets in a vice and hammer/bend them to fit your bike. Or add a stack of washers and use longer bolts to get the thing to sit right. You many also end up needing to add more bracing. With the kit you're looking at, note that they mention "spot weld the sprocket"... you will also need a welder.
I have a friend who has spent a lot of time on this, and eventually built a business around importing electric bike kits and bikes from China and Taiwan. His experience has been that since he's legally required to offer a minimum one year warranty, he has to use kits that cost him at least $500 and realistically more like $1000 (because people look at the "100 watts for 30 minutes" kit you get for $500 and don't see the point).