I think your big problem is that in the worst case, the characters on your number plates are only a few pixels wide, and the script really doesn't look ideal for OCR anyway.
By my calculations, at the top of each frame the resolution is around 6mm/pixel (assuming a 7m wide highway which is 1200 pixels wide at that point in the image).
A higher resolution camera sensor might help, but you may be limited more by the lens than by the sensor. A higher resolution camera might just give you more, blurrier pixels.
Before you go buying another camera, I would suggest that you need to determine what the limiting factor on image quality is.
Camera field of view
You may get significantly better results with three cameras, one per lane but zoomed in closer.
I would make sure that there is an overlap of half the width of a number plate between each cameras field of view, so a vehicle switching lanes would be captured on both cameras rather than a number plate being split between the two. You could stitch the images together after doing perspective correction, but that has other problems.
Assuming the same camera angle and resolution with a 2.3m lane and 0.3m overlap, that would increase your resolution to around 2.2mm/pixel (2.6m/1200) which is better than an expensive 4k camera trying to cover a 7m width (also, three HD cameras would be 3/4 the bandwidth of one 4k camera and 3/4 of the pixel processing).
Whether you orient the cameras in landscape or portrait will depend on resolution and field of view requirements, but also on traffic speed. While landscape maximises your resolution, portrait maximises the time dimension.
If you orient the camera in landscape and a vehicle travelling at the maximum expected speed can pass the field of view of the camera without the number plate being captured, then orienting the camera in portrait would increase the speed it would have to be travelling at to have a chance of avoid detection.
For instance, if your field of view is 2.6m wide and 1.46m deep and a car speeds past at 100mph (161kph, 45m/s), then at 30 frames per second (1.5m/frame), there is a (small) chance that only the top half of the number plate will be visible on one frame, while only the bottom half of the number plate will be visible on the next. If a vehicle is going faster than that, there is an increasing chance that it may not be in frame at all.
Reorienting the camera to portrait would cut the top of frame resolution from 2.2mm/pixel to 3.9mm/pixel (2.62600m/(1200/16*9)), but increase time resolution significantly, potentially giving you several frames containing the numberplate of even the fastest moving vehicles (assuming the shutter speed isn't too high enough that the individual frames are blurry).
I think before you spend more money, you need to characterise what you can achieve with the hardware you already have. If you can get good results with your current \$1000 camera when looking at a single lane, then your \$2000 budget could buy you two more identical cameras to extend the system to all three lanes (you may even be able to get a discount on buying a pair).
If this is a prototype, and you will be doing other installations elsewhere later, then you can look into cost reduction mechanisms once the prototype is performing well and you you have a test bed for trying out cheaper components.