I don't understand what you mean by "splat variables." Splatting is the act of feeding a hashtable to a command in order to provide parameter names and values. The way you're using & isn't splatting; it's invocation. And, have you just tried doing it?
The dollar sign is what cues the variable-replacement behavior in double quotes; precede the dollar sign with a backtick to override that behavior and not replace the variable. However, in your example, you can't do what you're asking. The use of single quotes turns off the variable replacement behavior, period. It seems like you might want:
Replacing the (backtick) placeholders with actual backticks. Those will escape the double quotes that they immediately precede, causing them to be treated as literals rather than as string terminators.