Oxidation concern (secondary)?

So I brewed my first lager (czech pilsner) two weeks ago, and after primary was completed and a 2 day diacetyl rest in my 6 gallon fermenter, I transferred to my secondary today. Unfortunately, I only ended up with about 4 gallons of beer transferred into my 5 gallon secondary. I am now going to lager at 35 degrees for 6 weeks. Because of the extra head space in my fermenter, I am concerned that I may have subjected my pilsner to oxidization. After lagering, I plan on transferring to my corny keg, and so at that point, I will be able to blast the oxygen out of the keg but am concerned about what might happen during the lagering phase. Thanks for any insight!

Do you have any CO2? You could shoot a blanket of CO2 on top of your beer to protect it. Or if you still have any active fermentation going on, you'll be fine. Just keep your secondary as still as possible so you don't mix O2 into your beer.

Why don't you just put it in the keg and lager it in the keg? I don't see the need for a carboy, if you've got a keg to lager the beer in.