The key here is to set up the vector so the stroke doesn't interfere with the integrity of the artwork. Start with a simple Copy/Paste in Place (
Command + C, Command + F) on the original image to preserve any details. Hide that top graphic (
Command + 3) and make the bottom graphic one block of color (same fill and add a stroke if it's not already there). Once you have the stroke to the desired thickness, make the hidden image visible again (
Command + Option + 3) to see results. If the stroke isn't where you want it to be, simply repeat hiding the preserved graphic and adjusting the stroke until you achieve the desired look you're going for.
That being said, please be sure to carefully check the license on the graphics from wherever you downloaded them to ensure that you can indeed make revisions or changes to the artwork.