What @wallyj said, but I also think any passenger will have more difficulty and lower priority when trying to upgrade on a codeshare.
Forget EzAir for a moment. If you are a UA FF, can you normally upgrade on a codeshare flying LH equipment? I would expect LH FF members are going to get first dibs on any available seats. Can you not book UA on the route?
EzAir won't be involved in airline upgrades, so don't bother asking them. You book the class you want and then it is up to airline rules and policy.
Sometimes, you will be able to either bid or purchase upgraded seat or cabin on airline website after booking thru EzAir. Again, it depends on airline - and ticket class I expect.
Bottom line, if you want something to happen, don't book a codeshare. Is it really that much cheaper? Often I find booking with the operating airline is less. I never book a codeshare unless no other choice or dollars are significant. You are often a second class citizen as a codeshare customer versus their own customers.