thing,
Bit of a broad statement to suggest that the OP is "bonkers" to fly a new type without flying it first with an instructor. Do we know what his overall experience is? What are the actual differences between the two types? What is the availability of instructors with relevant experience? What if the new type is a single seater?
In many cases some dual time would be wise or even strongly advisable; in other cases a ground briefing or even self briefing from the manual will be quite sufficient.
Many LSAs will have very similar handling characteristics to the Cessna and Piper types that flyer96 has already flown so the answer to his question is ...
Legally, your SEP rating qualifies you to fly LSAs so, if the insurance are happy to cover you then you can legally fly an LSA. It may be wise however to fly a familiarisation flight with an instructor or a pilot who has already got some experience on the type.
Happy landings
3 Pont