A logo has been passed on to me that was made with small dimensions (500px x 300px). It is not a vector logo and they use paint splatter brushes for the logo. So now they want the logo to be much larger and I cannot recreate the identical paint splatter effect. So I need to enlarge but the quality is being lost.
What is the best way to overcome this?
Here is a sample of the logo.
From what you've stated, you're a little stuck for choices. As it's impossible to add pixel quality, improving the quality is tricky.
Personally, in your situation I would print the logo as best as possible, then photography it as best as possible. Some photoshop editing, better quality. If you have a good printer and good camera, you should be good to go.