1) What working precision do you want for this calculation? The coefficients are ~ 50 digits, do you want 50 digits of accuracy? That will cause a bottleneck. Not to mention that the coefficients you end up with have over 3000 digits; this problem may need reformulating.
If you only require say double precision, then set Digits:=16: or if a lower target accuracy is required then set the digits lower.
2) If Maple is bottlenecking or runs out of memory, you need to make the input it receives as simple as possible. Try adding:
which will simplify the size of the input.
With these two changes, the first part of the calculation takes ~ 8 minutes with only 1GB of ram required on my old ThinkPad laptop. I am however using Maple 2015 as do not currently have access to Maple 18 on my home machine. (See attached, modified worksheet containing the first part of your code. N.B remove the convert(..,rational) command as further testing shows it lengthens the calculation time).
Another point is worth considering, but I did not use as it did not appear to work for your example, but may work for others.
3) In the Maple help page for invlaplace the following is printed:
"If the option opt is set to 'NO_INT', then the program will not resort to integration of the original problem if all other methods fail. This will increase the speed at which the transform will run."
I have spent some time testing this and I am unsure if this option is currently working in Maple, as I cannot find an example where it offers any speed increase over not using it. This could be worth considering though.
I do not currently have time to look at the second part of the code, but will see if I can revisit it later. You may want to look into the gc() and unassign() Maple commands in order to free up memory.