BRP delay, Tier 2 approved - any success stories of re-entering without BRP?

Hi all, my in-country Tier 2 change of employment visa was approved last Monday. I was told I would receive the BRP within 7 days, which have lapsed (it's now Wednesday). The BRP delivery delay is making me anxious as I have urgent family travel. I have a non-EU passport that allows me to use the egates. What are the risks of travelling without the BRP if I can show the letter that says my visa has been approved? Has anyone been in a similar situation - able to re-enter problem-free or if stopped by immigration able to explain and let in? Thank you.

It is 7-10 WORKING days for the BRP.

Phone DX Couriers to ask if they have a delivery for you.

You should wait to get your BRP before traveling. Being admitted to the UK as a visitor will cancel your tier 2 visa.

Assuming I am successful entering through the e-gates, how will they distinguish if I am re-entering as a tourist or a resident?