I constructed my kayak setup with what you want for within your budget. All new too. The problem is, it was a LOT of DIY work on my end. I can't say how long it took to get it all together, because I did it little by little, but I added it all up once and it came in at around $1,300, probably not including taxes. Plus, prices have inflated since then, so going the same route I did will probably still blow you budget. But I still believe it's going to be the cheapest route.
And not everyone who fishes from a kayak winds up with a trolling motor. A lot of people prefer pedals over trolling motors because unless you get a really expensive trolling motor setup, like one with spotlock and a remote, you'll usually have more control with a pedal kayak.
My kayak is a Lifetime Yukon. It was a really good deal when I got it (I think I paid $450 for it) though its price has gone up a good bit since I bought mine (I think around $600-700 now). Then I installed a trolling motor on the stern, which was VERY difficult because there's no access to the underside of the kayak back there. And you have to reinforce the mount from underneath to keep it supported so it doesn't break or tear out the plastic. AND I had to build the mount and reinforcement pieces myself, as there's not an aftermarket solution for this kayak. Then I took a Minn Kota Edura C2 30 and hacked the top off and built a PWM speed controller. Then I removed the original foot pedals, installed new ones (with controls), and installed teflon sleeves and SS wires through the hull (again with no inside access) so I could control the trolling motor. My point being, it's doable, but if you don't buy a kayak specifically designed for a trolling motor (which tend to be pretty expensive), it's not going to be easy to do yourself. But, it's not impossible either.
So, if you love making stuff with your hands as much as I do, then there are options. And just about any kayak can be retrofitted with a trolling motor. But, if putting in a ton of work doesn't appeal to you, then I'd recommend saving up your money and buying a kayak that comes with a trolling motor already built in. However, the other good thing about DIYing a trolling motor is that you can buy the kayak now, fish out of it, and then save up more money to install the motor at a later date.
Installing a fish finder usually isn't too hard. There are tons of aftermarket add-ons that are fairly cheap and make the process pretty easy.
Though really, if you can find a used one, that's the way to go. However, I suspicion that it won't be easy to find a used kayak with a trolling motor already built in. I don't see too many on the lakes I visit and rarely see one for sale for less than about $4k. Though, they're typically high-end setups with tons of add-ons included in that price, so they're still a significant savings over new. They're just not cheap. But hey, I'm sure they come up for sale from time to time, so if you're patient and keep vigilant long enough, one or two is bound to pop up occasionally.