The OBi speed dial has two advantages: the phone will be sending only two characters, e.g. 2#, rather than 7, 10, or 11 digits depending on your dialing format. Also, unless your digit map is carefully set up, if the number does not end in #, the OBi will wait an additional delay to see whether more digits are coming.
If your phone has dedicated buttons that function as speed dials, it makes sense to use both -- set a phone button to send 2# to the OBi, with the OBi set up to dial the station when 2# is entered.
Using Gmail won't solve the early media problem -- I only suggested it as a way to make the call by pressing only one button.
If you want to avoid the early media issue, you will have to use a non-free service. That's won't cost much, because nearly all your calls will get a busy and so won't get charged. However, unless you have such an account for 911, as a backup, or for some other purpose, it probably wouldn't be worth the hassle to get one just for contest calls.