@Elizabeth Naranjo There is no easy way out. Most areas require a permit to do as little work as installing 1/2 a sheet of drywall. Certainly replacing a furnace and/or a water heater would require a permit as well as running a new or replacing a duct run. Adding a new outlet, circuit and pretty much doing anything electrical aside from replacing a fixture or replacing the receptacles or switches requires an electrical permit. For plumbing, any new piping (even replacing existing pipe with new) would require a permit. Moving a gas appliance (range, or furnace) would require a permit. Some areas require permits for windows and some do not. Replacing a roof would require permits and sometimes replacing or installing new siding requires permits.
Usually plan on permits unless it's a paint and carpet rehab.
I think you may be over estimating the time required to obtain permits. Most permits for single family residential rehab type work is done over the counter. The exception being if you are changing the floor plan which may mean you need architect drawings. If you are changing the location of fixtures they may require a plumbing plan as well. It really depends on the scope of work and the person inspecting the work as well as the one issuing the permit.
Find someone who knows the permitting system in the municipality where your project is located and have them review the scope, and the work done and see what they think it would take to pull permits after the fact. What I'm saying is you might be over estimating the time and difficulty of getting the project completed with permits. If the work was done properly, it may not be too much of a problem. Fall on your sword with the municipality and tell them you were taken advantage of and want to move forward correctly now. Most are somewhat understanding and most will try to do what they can to get your project finished. As long as you don't argue with them or make them mad, you should not experience more than the usual brain damage of getting permits. Sure it will cost more and take longer but consider it the price of your education.
Other than that your option as others have point out is to find the money to finish it off the books of the hard money lender, sell it or refi it and pay off the lender and move on. I would say that about 98% of all residential rehabs in the Denver market are done without permits. Occasionally you will see listings in the MLS that state "all work done with permits" or something similar but in most cases, it's a don't ask, don't tell approach. That said, that now that the market has softened a bit, folks wanting the work permitted is probably going to be more common so it may not be a bad thing.
My final thoughts are about your hard money lender. My assumption is that this is not their first rodeo and as such, their loan documents should have covered the need for permits for partial payments on the improvement amounts.
I think you realize now that you hired a GC that lied to you. I would have serious reservations moving forward with this person to complete your project. They should have clearly communicated to you what work required permits and that they could get those or not but the choice would be yours and that you would bear the cost implications either way. Since they did not do that, what else are they hiding from you? Sub-par work, cost over-runs, undisclosed delays and work beyond their expertise? You need to slow down and get your team on the same page with you or it will cost you more money and time in the long run.