To get a laptop with any chance of rendering a decent quality scene would cost at least twice your budget.
Rendering in Iray uses your CPU and/or your Nvidia powered card's CUDA cores. Laptops already have underpowered CPUs because of cooling constraints, and Nvidia's best laptop GPU has 384 CUDA cores to play with, while the "entry level" GTX 1050 for desktops has 640 CUDA cores.
Then there's the memory. Laptops don't have dedicated GPU memory, so unless you fork out even more for 64GB RAM, you'll run into problems with more than two characters on screen, and that's ignoring that the memory on graphics cards runs much faster than regular memory.
To give you an example of Daz performance, I have a 1070 ti with 2,423 CUDA cores and 8GB memory, and the attached scene took roughly half an hour to render.
To get that quality on a $2000 laptop would still take 5x the rendering time at least.
So yeah, save yourself a headache and a lot of money and go desktop.
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This statement about memory simply isn't true UNLESS you only have an integrated GPU, which by definition WON'T be an Nvidia GPU. Yeah Intel did a few Intel-Radeon hybrid chips, but most Intel CPUs have Intel integrated graphics. And Radeon GPUs and integrated GPUs are useless for Iray renders, although you can still use them to drive the viewport.
Laptops with discrete Nvidia GPUs have GPU memory included with the GPU, which is separate from the system memory. While my laptop is WAAAAAAY out of the OP's budget, I have two GTX 1080s with 8 GB of GPU memory each, PLUS 64 GB of system memory (all that system memory is major overkill for what I use this laptop for).
And, my Iray GPU benchmarks come in with about the same numbers as the desktop variants of dual 1080s. The desktop GTX 1080 variant might be a hair faster, but not noticeably so. My particular laptop is better defined as a desktop replacement, though, due to it's size.
Of course, with a desktop, you have the option to upgrade the GPU to a 1080 Ti or better at a later date, or maybe add another GPU card if you have a spare PCIe16 slot and suitable overhead on your power supply, while with a laptop, yeah people generally don't upgrade laptop GPUs (it is possible with certain laptops, but usually this just isn't an option). There is the external enclosure option with a Laptop if you have a Thunderbolt port though, but then those enclosures aren't really portable, as they generally don't have their own battery supply. Plus that's more stuff to carry around...
Per discussions on the Daz forum, the trick is to find Nvidia GPU laptop graphics with the most memory. If the OP can stretch his budget slightly (to $1200 or so), and watch for sales, you can get GTX 1060 graphics, which has 6 GB of onboard GPU memory.
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There are cheaper options with 4GB or less of graphic memory, but it's pretty easy to exceed that 4 GB, especially with Windows reserving around 18% of said 4 GB for itself, hence leaving you maybe 3.2 GB or so to work with.
The other issue with laptops is cooling while rendering. GPU throttling can help with this, but then of course your render times will be higher. Still MUCH faster than CPU only renders though. And rendering sucks battery life, but of course you can build scenes while you are on the go, leaving the full on rendering for later when you are able to plug in to a nearby AC outlet.
And if you aren't doing Iray (i.e. are doing 3Delight instead), then the graphics card isn't going to matter much, as you'll be doing CPU only renders anyways. Most people like Iray though...
There's the whole utilizing a 'render farm/cloud rendering' thing as a possibility too, where you build the scene on your system, then use some render service to do the render for you, but those services aren't free. Plus there's the question about using a remote service to render questionable content...
In short, it's possible to go the laptop route, but normally it's recommended to go the desktop route, unless your usage pattern favors laptops heavily.
