Multibit developers have no access to your funds so I'm not sure what do you mean by "prove" that you had bitcoins. If you have the .key though, you should be able to open it using any text editor since It's a plaintext file and then, just take the private keys and sweep them to your Electrum wallet. If there is a password on the file, import the .key first and then export it again without the password.
I need to make a small transaction from my old wallet in order to prove it is mine.
If I sweep the keys in Electrum and then make a transaction I'll still have the same address, right? I have the password and all info. I realise I'm probably being really dense but I don't want to mess up anything here... thanks.