Is this scheme for a single-fan electric airship with a rocket-like geometry workable? I am currently designing a vertical take-off and landing vehicle that uses a Tesla powertrain connected to an electric ducted fan. Please comment on if my thought process is sound on my feasibility analysis.

My initial mission goal is extremely modest to grease the wheels of my feasibility analysis and have a realistic utility in society (and for people in my sport, recreational/experimental skydiving).

The vehicle simply needs to lift itself and a 100kg payload (coincidentally the approximate mass of a skydiver with his parachute) to the altitude of 1 mile.

I think I need to calculate my total delta_v, but because I don't need a high speed (like an orbital escape velocity or something like that) I am not positive how to go about getting it without first understanding the maximum thrust from my system.

My current plan for determining thrust from the tesla-system: 1) identify maximum output of tesla battery and motor (lets say Model S P100D) 2) calculate maximum propeller size for #1 based on algorithms available in electric drone aircraft websites/forums 3) calculate potential thrust produced by #2 4) use #3 to determine if liftoff is possible carrying at least #1

My initial calculations are not intended to include all outside forces; I will incorporate these if it is in fact, theoretically feasible at all.

Based on information above as well as the weights of the battery, motor, and an estimate of a basic rocket-shaped frame, I can figure out if a basic propeller will work for this, or if I will need to incorporate an electric turbojet, which would blow up my budget for this.

It is also feasible I could have a combination of several motors / batteries attached to multiple propellers with something like a rocket sitting in the middle of an oversized quadcopter. Again, budget goes up considerably.
The best propeller driven vertical take-off and landing craft would be a helicopter, actually. Yes a duct around a propeller increases thrust, but upsizing a propeller creates way more thrust from the available power.

Propeller thrust has scaling effects with size due to different Reynold numbers, data does not scale up well. Best way forward from where you are now would be to look at the weight, rotor dimensions and installed power of existing designs for small helicopters such as the R-22. Or better yet, design an electrical version of the Hiller Flying Platform Platform.