I use BOTH LogMeIn Pro and Dropbox. I find them both to be invaluable.
Another advantage of the Pro account over the free account is the ability to remote print. If I am logged onto a clients machine with a Pro account, I can print a copy of the file I am working on to my computer. This comes in handy if you are preforming accounting functions and need to print checks locally.
To overcome the lack of file transfer in the free account you use the LogMeIn free account in conjunction with Dropbox. You just have an extra step to transfer a file utilizing this method. You would need to access the client's computer with LogMeIn then place the file you want to transfer into the client's Dropbox within a folder that is shared with your Dropbox. Then you can access the file on your computer(s).
Best of Luck