When are transition state's energy barrier "reasonable" at a certain temperature?
Your question is really about what you consider to be a reasonable rate constant and so is somewhat subjective. The fastest a bimolecular reaction can be in solution is given by how fast the reactants can collide and then react on first collision. i.e reaction is diffusion controlled. In this case as reaction is at first meeting the activation energy must be small, $\le \pu{5  kJ mol-1}$. In water the diffusion controlled rate constant is in the region of a few times $\pu{10^10 dm3 mol-1 s-1}$. Most reactions are far slower than this. 

The simplest, but very approximate, approach for other reactions is to use the Arrhenius equation to estimate the rate constant. The equation is 
$$ k = A \exp(-E_a/RT),$$ 
where $E_a$ can be taken as your activation energy and you have then to estimate $A$. This can  be done for simple reactions using statistical mechanics and calculating the partition functions, but for anything complex you are left only with guessing values based on experimental data from similar reactions. However, because of the exponential dependence of the rate constant on activation energy even a small change in this can have a dramatic influence on the rate constant, far bigger than any similar change in $A$. 

If you do have a potential energy surface then you can calculate trajectories along this surface by solving the (classical) equations of motion, repeatedly using energy sampled from a Boltzmann energy distribution,  and so work out a theoretical rate constant.