What's going to matter is how the banks report this transaction to the IRS. Have you completed this type of transfer/rollover in any previous years?
If HSA Bank considers it a distribution (there's no distinction on this end between types of distribution even if it's subsequently rolled over) then your 1099SA will include the distribution in box 1. On the other end, Elements will add a rollover contribution to box 4 of your 5498SA (but this won't come out until May of the following year).
I've never done a trustee to trustee transfer because my HSAs have always charged for that, but
https://www.irs.gov/instructions/i1099sa says that they shouldn't be reported on 1099SA, so if you don't see anything on a 1099SA from HSA Bank for a year in which you did the transfer then you should be good.
If this is the first year you're doing this then I'd probably just do one this year to be safe and check how it's reported. I wouldn't rely on anything customer service tells you either way as I've generally found they're not reliable on technical/tax issues.