What is the extent of the effect of Tris on E. coli?
The cited article describes effects of Tris on the outer membrane of E. coli. Like all Gram negatives E. coli has an inner (or cytoplasmic) membrane which is a typical phospholipid bilayer membrane containing many membrane proteins including transporters and, of course, an electron transport chain. Surrounding this there is a layer of peptidoglycan, and then the outer layer of the cell envelope is the outer membrane, which is a special (highly atypical) bilayer structure containing phospholipid and lipopolysaccharide as well as a number of proteins including the porins. The space between the two membranes is the periplasm, or periplasmic space which contains a number of small secreted proteins.

The outer membrane is susceptible to disruption by a number of treatments including EDTA treatment. For example, many cell lysis procedures for E. coli use EDTA to disrupt the outer membrane sufficiently to allow lysozyme to gain access to the peptidoglycan.

Cells with destabilised outer membranes could display increased fragility, but in general I would imagine that this would not normally cause any specific problems: obviously this would depend upon exactly what your experiments are.

Finally it is worth noting that the concentration of Tris that you will be adding with your kanamycin must be pretty low. Even if you have used 0.1 M Tris you are presumably using a 1000x stock, so the final concentration of Tris will only be 0.1 mM. The work in the cited paper looked at cells resuspended directly in 0.1 M Tris. I don't think that you need to worry about this at all.