If this was a normal pan I'd be scraping it with a metal spatula right now, but it's a custom shaped pan (for making novelty shape bread/cupcakes). There are no flat spots, just a lot of grooves and curves. Should I be using a plastic or otherwise non-scratchy tool to scrape out one part at a time, or is there some other approach?
You should not use metal on cast iron or may damage the seasoning. It is best to clean it while still hot with a brush to get into the grooves and never use dish soap. Some coarse salt can help get rid of burnt on areas. Then reheat and coat with a bit of oil to protect the coating/seasoning. Patterned cast iron is difficult to season and may take more time developing a good seasoning.
As mentioned in the comments, start with a short hot soak (the hotter the water, the better, even boiling water poured into the sink is great for this). Remove the pan from the still hot water and scrub off all of the stuck-on stuff that you can using a nylon brush or a scrubby sponge . Repeat with fresh hot water as necessary until all of the big stuck-on stuff is gone. Once you get down to just little stuff, use kosher or other coarse salt to scrub it off. Do not use soap, do not use steel wool or a metal brush. Rinse very thoroughly.
NOW is when your pan is vulnerable to rust. Get it dry fast by using towel and paper towels, then stick it a hot oven (or on a burner) just until the whole pan is warm and completely dry. Lightly oil the whole pan (I use a paper towel for this), bake the oiled pan upside down at 350F (180C) for an hour. Once cool, it's ready to put away.
Get it nice and hot, nearly red hot, the stuck things will seperate and then knock it a bit to loosen anything left. But for safety, use tongs and oven mits/kitchen towels to handle it while it's hot and make sure you have some place to put it when you take it off the burner, like the sink or ceramic/stone surface, that won't melt or scorch. And as Jolene has mentioned, a short soak with a thorough drying and re-oiling should be fine for the finer grit.