Jungfraujoch in mid-September — buy winter coat or rent in Interlaken?

Good day all.

I'll be visiting Switzerland in mid-September and Jungfraujoch top is a top priority on my itinerary. I'm coming from a hot country and don't currently own any winter clothing/ shoe.

A few questions:

1. What should I realistically expect temperature/weather-wise at the top in mid September?
2. What layers/clothing would be appropriate for the visit (I will only be at the top for 1 day with outdoor activities)?
3. Given I'm doing a semi-backpacking trip and want to pack light, is it worth buying a proper winter coat and bringing it along, or would renting gear in Interlaken make more sense as I will only go to the top for 1 day?
4. If renting is the way to go, does anyone have recommendations for rental shops in Interlaken (or Lauterbrunnen) — price ranges, what's typically included, etc.?

Would appreciate any first-hand experience, especially from people who've done Jungfraujoch around the same time of year. Thanks in advance!