Handy knowing the budget, makes for an informed choice or recommendation
Personally I'd suggest something like a longtail cargo bike. Keeps the youngster with you, rather than being towed behind. I think the pull along wheeled chariot is pretty antiquated these days as theres a lot of people swopping over to ebike for shortish journeys and the choice available now is far better.
Alternative to the longtail is the box in front type, favoured by parents in Europe.
Kids sit in the front, easy to keep an eye on.
For an example, look at manufacturers like Riese and Muller or Tern, though these are premium brands and its just for an example.
Also take a look at cargo bikes in general, give yourself an idea of what they can offer.
Kits are fine if you understand the basics of electronics and bike mechanics and are prepared that there might be issues or a bit of problem solving to deal with.
Off the shelf Ebikes you always have the comfort of the guarantee.
My own thoughts on the kid trailer is i really dont like the concept. I've seen too many car accidents of cars being rear ended, I would not like to think how bad it could get should it happen with a kid trailer.