Advice: Try to keep track of all the constraints/goals and your design decisions. Aim for a balance, and try to keep track of the impact of your changes. I often create a mind-map to capture the scope of all the components/subsystems, or some 2D drawing, so that I can see 'everything' and their relationships in one glance.
Making the robot as light as practical should reduce the cost and complexity of the robot's motion system.
Getting the cameras off the robot would also allow yourself to use a lot of cheap, heavy computer power, so that seems like a good idea.
If the cameras can see the robot and the projectile, it probably only needs to know the robots orientation, so that the robot can be driven in a direction to avoid the projectile. Robot orientation might be doable if the camera(s) see different coloured LEDs on the robots (visible) corners.
BLDC's have a lot of torque once under adequate control. However they are more complex to drive than simple DC motors. In my (limited) experience, getting BLDCs to start is the hardest part, which would be a disadvantage in your application.
For a fast start, the motor drive needs some feedback to know their position. I have some off the shelf BLDC controllers, intended for use in models. They take take a noticeable fraction of a second to get the BLDCs moving, and I have seen this to be the case for several other combinations of BLDC and motor control.
With a limited budget, where you may only be able to afford the cost of one set of parts, I strongly recommend you do plenty of investigation before trying BLDCs. Personally, I think they may be a bit too expensive for your budget; my low-cost BLDCs with controllers (supporting reverse, many BLDC controllers don't do reverse, but I suspect your robot will need it) were over $40.
If the robot can be reduced in weight to a few hundred grams, then small geared DC motors are powerful, popular, modest cost and easy to drive. E.g. 12mm gear motors which are available from many suppliers, though the quality of the very cheapest can be different from these (they can be had for about $6 each). You may be able to afford 4 of those. I have seen small 4WD robots (150gm) using them with off the shelf 'wide wheels', and they 'wheeley' because there is some much traction and torque.
Folks who do 'Ant Weight' and 'Mini Sumo' robot wars use those DC geared motors. They come in a range of gear ratios so you could aim for a balance between speed and torque. They need a motor controller, but there are plenty of ready made controllers from internet resellers for a few GBP capable of controlling more than enough power.
Those robot wars folks also use model servos. Again they come in a range of power and speed, can be modified for greater than 180 degree rotation, and have their own motor control electronics, and so might be quick to get working.
As well as the motors, you should think about the cost of powering them. LiPo batteries have the greatest power/weight. However, you may end up spending 30GBP+ for a battery charger on top of the cost of batteries. NiMh are not as good. However, AA or AAA are easy to buy, and the charger is cheap.