some ideas to consider:
First, simplest, make a number of named Export presets, each with its different naming and resizing options, and then just run through those in turn, on a given highlighted batch of images.
Next way might be to record a new Photoshop action, which performs a number of sequential resizes (plus any standardised corrective sharpening) with separate dedicated saves along the way, to the current image. You'd start with the highest resolution you were going to be outputting, and simplify down progressively from there. The extra degradation (compared with each version being made directly from a primary image) would just need to be accepted, very slight as it would be, if you wanted the simplicity of this.
Then embed this PS action into a "Droplet" and add this Droplet to LR's export processing folder.
In your LR export setup (I'd save this into an Export preset, for repeatability) you'd configure the Export as you wanted it to be, at the beginning of your PS Action's sequence. Then you'd select the Droplet in the Postprocessing section of the LR Export dialog.
Running this Export for a batch of images would, for each image in turn, make that starting version and then invoke PS, and tell PS to run through the previously created Action. Then move onto the next image.
Another entirely separate method might be to invoke something like ImageMagick in your Export postprocessing - it would need a very long and complicated command line full of options, to tell it what to do, but may be doable and should be swift. Not involving PS.
Thanks for your help Richard.
I have got presets already done but want to make it easier. I've made an action in Ps which works 'per image' but says 'image size is not currently available' when I batch process with the same action. I think I need another part of this forum.