I realize this question was posted a while ago, but I would like to make a note on your edit stating,

After looking around some more I found time scales, which are pretty much what I was thinking of in the second part of my question (though many of the answers people have provided are along the same general lines). I'm surprised I don't hear more about this in analysis - unifying discrete and continuous should make it a pretty fundamental concept!

Time scale calculus is fairly new development. It came about in 1989 in Hilger's Ph.D. thesis (he initially called it a measure chain). At the time of writing his thesis, mathematicians did not embrace his idea. Then, in the early 90s, the ideas were brought to the U.S. and now there are papers being written about time scale all over the world. However, I suppose there are still some mathematicians who are not studying/teaching time scale.

If you are looking for resources on this, you might consider taking a look at [1].

Furthermore, I see above that you are trying to define the exponential function for "discrete(difference) calculus" (if I am understanding your posting correctly).

Note that for the continuous case, we have that $(e^{at})^{'}=ae^{at}$

Now, consider $\Delta (1+ \alpha)^{t} = (1+ \alpha)^{t+1} - (1 + \alpha)^{t}= (1+\alpha)(1+\alpha)^{t}- (1+\alpha)^{t} = ((1+\alpha)-1)(1+\alpha)^{t}=\alpha(1+\alpha)^{t}$.

Thus, the discrete exponential should actually be $(1+ \alpha)^{t}$.

[2] is a good resource for discrete calculus.

Also, [1] discusses exponentials in time scale. There is a delta and nabla exponential form. You may consider checking that out.

Now, to your fractional calculus question, I believe I recall my professor mentioning some problem with fractional calculus and time scale. I cannot remember what it was exactly, but it was an open problem regarding fractional calculus. (Note: There are many open problems in time scale.)

I hope this helps!

[1] Bohner, M. & Peterson, A. (2001). Dynamic Equations on Time Scales: An Introduction with Applicaitons. Boston, MA: Birkhäuser.

[2] Kelley, W. & Peterson, A. (2001). Difference Equations: An Introduction with Applications (2nd Ed.). San Diego, CA: Academic Press.