Okay, there's a couple options I can think of.
One possibility is simply, to cool down your oven. You're increasing the baking time anyway, a cooler oven will let the cake cook more slowly but more evenly - it is a classic way of not overcooking the outside of a large dish before the inside is done.
Another possibility is, you watch your cake a little more closely, and interfere when you notice any problems - for example, tenting foil over parts that are browning too soon, or adding moisture by the eyedropperfuls if the surface (or edges) are looking a bit too dry. You can also turn down the heat more, or turn the heat off and let it finish in residual heat if nearly done, if the outside is browning faster than the inside is cooking. This will take a bit more time, effort, and mindfulness, since you have to be there and paying attention for a fair chunk of the cooking time - but it is also the most generalizable advice for baking in odd-shaped pans.
Last possibility is, you can pad your pan before baking, focusing on any trouble spots. You mentioned double lining the pan, and while it might have helped - it clearly didn't prevent the problem, so why not try doing more next time? One historical technique was to wrap the outside of the pan with paper - newspaper, brown paper, or parchment paper, a few layers (and the air trapped between them) would slow down browning on the outside and give the interior a bit more time to come up to temperature. Or you can use cardboard for the same purpose.
Or else you can use a more durable equivalent and try a silicon mat (plus or minus strips for targeted protection), or make up something using cheesecloth, or repurpose an oven-safe ceramic or glass lid or pan to shield known trouble spots, or something of the sort. You could also make something similar to a crust shield, like is used on pies - which would use strips of foil along the edges to prevent them from overbaking while the center finishes cooking. You wouldn't find pre-made ones like there are for pie, but you can use the same principles to position a few strips of foil over the tops of trouble spots to let them cook slower.
You would want extra padding where you expect problems (in this case, a few extra layers around the top curves). This setup would shield the outer layers from radiant heat a bit more, and let the batter warm and cook a bit more slowly but also more evenly. You might not want newspaper if you're putting any layers over the top of the cake pan, or where it might touch the cake, but it should work for just padding the pan with.
Clearly, this third option works best if you know, or can guess, the trouble spots beforehand. Perhaps the first time baking in an oddly-shaped pan, it would be wiser to watch over it and meddle in person, and subsequently you can know where trouble is likely to develop, and can preemptively pad the pan or shield the cake to compensate.