Welcome MariaC,

I don't know of any wooden-handled strops, but that doesn't mean they don't exist. In your budget you could get a nice brush with wooden handle - those are quite popular and come in all kinds of styles - from pieces of wood in their natural shape, to turned on lathe, to stacked from multiple woods and other materials, to carved...
There are also wooden handles for safety razors - basically you screw the head of the razor on to it.
Wooden razor scales are also popular, but that's either not going to be complete razor or over budget.