Is there a GRAMMAR mistake.... There's a spelling mistake in the question. "I suggest our starting as soon as possible," is a rather colloquial/informal way of saying it. Some people here in England do say it that way. I've said it myself! I suggest that you stick to the other examples you have mentioned. They are better.

You are right. As you say, there is a grammar mistake in that sentence. (NB note my correction of your question). Either 'I suggest we start..' or 'I suggest starting..' would be correct.

"I suggest we start as soon as possible" would sound the most natural.