GG, I would not worry too much about your tree. Crepe myrtles are extremely hardy. Aphids are particularly prevalent this year. If it.were me in your shoes, I would not move the tree due to aphids. It probably wont protect your tomatoes from aphids and it will set your tree back.
Fyi The Crepe Myrtle is going to be a handful to move the roots spred out in a horizontal way so you are going to need lots of help ..About the sprayer I put one of those brass nozzles on mine the kind that comes to a small opening and that lets me spray to the top of any of my fruit trees,,Good Luck on the Aphids
In a garden project that I am doing for the city (as a volunteer ) we had 4 crepes to deal with and the day we started work on the project it rained so getting a backhoe was out of the question so we just pruned them severely and they came back quiockly but made nice small shrubs and are now blooming happily adding an accidental bright red border...just saying that they can be managed just like any flowering plant ...BTW we cut the to ground level
I do pretty much the same thing with the Black Locust it is the worst weed/tree here on my small city lot any time that you dig for any reason you will encounter tree roots and most will be the Black Locust and any root that is cut will produce a new tree and will grow 12 or more feet in a year I cut them off and drill a hole down into the stump and fill with something a little more toxic than venegar..
We're undergoing an Aphid invasion around Austin, especially the Pecan trees followed by Crape Myrtles; folks are complaining about falling leaves sticking to their shoes and sticky gunk covering cars, sidewalks and house siding.
Concerned homeowners with large Pecan trees are coughing up $700+ per tree to get them "sprayed"! In close quarters I'd be concerned about spray drift carrying chemical insecticide into my yard and onto vegetables and herbs in my garden!
Wizzie Brown (Travis County Agri-Life Extension Entomologist) suggest least toxic action first: concentrated water spray (mite blaster wand), insecticidal soap sprayed late in the afternoon (soap burns leaves in heat), Neem spray from concentrate - NOT Neem oil. All this has to be directed to the underside of the leaf and repeated to be effective but does nothing for the honeydew coating leaves, and then there's the problem of foliage being out of reach.
Although the Aphids are causing havoc now, their population will crash at some point - and - using chemical sprays may knock them down only to have the population rebound.
If natural populations of predictors were in balance we probably wouldn't notice the Aphids at all!