I have a hard time with injecting glyphosate into hollow stems for treatment. I've read this as well but the way glyphosate works is to be taken up by the leaves, vigorously growing leaves and it is incorporated into the physiology of the plant's chemical processes to mess with this 'enzyme' without which the plant dies. Taken into the plant this way takes the glyphosate down to the roots. Cutting down the Knotweed should be done 3 weeks after spraying.
Glyphosate: mechanism of action. Glyphosate is a herbicide used in
agriculture and non-crop situations for the control of a wide range of
weeds. ... Once absorbed by the plant, glyphosate binds to and blocks
the activity of the enzyme enolpyruvylshikimate-3-phosphate synthase
(EPSPS).Jun 19, 2013 Glyphosate | Glyphosate: mechanism of action
To fix your problem, spray at night using large drops not a fine spray. These plants are usually ONE mother plant. Literally. Just spray some of the leaves here and there. Glyphosate will be taken up and transferred throughout the plant including the roots. Chopping off the bulk of the photosynthesizing plant factories doesn't make sense at all. Going through the photosynthesis process is how glyphosate works to become systemic. It doesn't work at all by contact like hot water, vinegar and acids. I have read other articles that tout this method. I would spray at night using a sprayer you can control the droplet size. It will dry before the butterflies and bees become active. Wait 3 weeks then cut them down and inject with a big fat syringe into the canes.
Be careful cutting the plant down. Don't use a line trimmer/weed wacker. If you can, use loppers with long handles. Keep a tarp next to you to put the debris on as you gather the canes and foliage, raking up the dead stuff already on the ground carefully. Any part of the plant has the ability to regrow.
You'll be doing this once a year for a couple of years. You are essentially starving the Mother plant as well as killing the roots with glyphosate. Disturb the soil as little as possible. Pulling out roots leaves too much chance of breaking the roots starting a new 'Mother'.
Consider dumping soil or compost or bark chips on top of the soil about a foot or more deep.