You can call a postprocessing program as part of an Export. So for example, you might export your high res TIFF as the prime requirement from LR, but in the same Export settings which make that happen, also point to an external utility such as "ImageMagick" - including as you do this command-line instructions on what should be done by that - such as, generating and saving other derived versions working from this particular exported image.
Another postprocessing option would be to encapsulate a Photoshop Action (such as, one which resizes and saves in a certain way) into a "droplet" utility file and then call that from your LR Export. The droplet will start up PS as required and then script PS to carry out the needed tasks straight away; the Action concerned would need to include for a Save IIRC.
A different approach might be to define your various differently sized and saved export requirements into separate Publish setups. Then within each one, maintain a publish collection or Smart collection which picks up the images concerned - e.g., images edited within the last week that have a certain star rating. You would still need to click "go" for each different publish service, to make its own particular sort of output happen - as needed; it will not repeat images that are unaltered since last time - having accumulated a batch requiring this.
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