There is an amusing engineering expression: 'A Camel is a horse designed by a committee'.
In engineering a mechanical assembly, you should start with objectives and parameters and then design to satisfy your targets. For example, you don't make a soup spoon out of titanium just because titanium is 'better'. Running hot is not automatically bad, either. Heat is actually beneficial in many cases. Just don't exceed the specs.
If there is an unlimited budget, then the smart move here it to simply plunk down the cash and purchase a high speed spindle, electric or air, as desired. However, the primary parameter here has been stated by the 'customer' as cost, so living with the tolerances offered by 'cost effective' solutions is simply a necessary part of this particular project. It is unrealistic to expect .000000000001" runout for 25 bucks.
A good example is the heat issue. Aluminum is obviously better at getting rid of the heat, so you would want to make the body from aluminum . . . but only if you can tolerate the accuracy loss generated by the clearance you would need to design into the bearings to accommodate the expansion difference between the aluminum casing and the steel shaft within. If you are using high clearance ball bearings, probably not a problem over 4" distance. On the other hand, if you have a precision angular contact set, the expansion difference will erase your pre load . . not good.
Bearings have a temp spec. Hybrid ceramics can take a lot more heat and are not nearly as expensive as full ceramic. Simply using a cutoff tool to make . . oh, say 20 or so radial grooves .100" x.100" cut into the casing OD may provide adequate cooling for hybrid ceramics, which run hot, but thats perfectly OK. Clamping aluminum mounting blocks right at the bearing sites will draw off a great deal of heat as well. Then you could get fancy and cut cooling fins in the mounting blocks . . etc, etc.
A left a question unanswered some time back, so here is the answer: For my previous little X2 mill, in order to run the spindle at 7,500 RPM, I simply attached an old Pentium 3 aluminum heat sink to the side of the head. That was enough to keep the beargins in spec. Low tech, cheap, ugly, but completely effective.