Where is it located, exterior wall access where you could punch through? What part of the world are you in? Fart fans are not made for static pressure so when you shove a bunch of air into a small 3in duct it can't push it. It would be best to have a decent fan and 4in duct but if that's not possible downsize the fan. What fan did you install? What size is the bathroom?

What's the original build date of the house? That would tell us what kind of conductors could be back there. That taped splice is a big issue. All connections have to be made inside a junction box with approved connectors (wire nuts or wago type).