Does such a word even exist? If not, what should I use as an alternative?
"Hobson's choice" may be appropriate, ie. the outcome is utterly inevitable despite the illusion of choice, the allusion being to one Thomas Hobson who had a (ie. just one) horse for sale, in spite of the appearance that he gave to his customers of having more than 40: https://en.wikipedia.org/wiki/Hobson%27s_choice
As to if one word exists to fulfill the requirement, beyond words such as inevitable/inevitability, and phrases such as convergent-evolution, I'm realy not sure anything directly fits. I should say that perhaps doom/doomed is the closest but has rather sinister overtones in modern English. Ordained, predestined, predetermined, others - but still none reference choice in the way the Hobson's choice does. https://www.merriam-webster.com/dictionary/doom
Not a single word but a common saying is:
said to mean that all the methods of doing something will achieve the same result in the end.
It might be a Catch-22.
There was only one catch and that was Catch-22, which specified that a concern for one's own safety in the face of dangers that were real and immediate was the process of a rational mind. Orr was crazy and could be grounded. All he had to do was ask; and as soon as he did, he would no longer be crazy and would have to fly more missions. Orr would be crazy to fly more missions and sane if he didn't, but if he was sane, he had to fly them. If he flew them, he was crazy and didn't have to; but if he didn't want to, he was sane and had to. Yossarian was moved very deeply by the absolute simplicity of this clause of Catch-22 and let out a respectful whistle.
It’s used in situations where there appears to be a choice but because of rules and regulations, there isn’t actually a choice, therefore all outcomes are identical regardless of any choices made.
Predestination means that the outcome has already been decided, but the dictionaries are rather theological about their definitions.
the belief that people have no control over events because God or another force has arranged everything that is going to happen
the belief that people have no control over events because these things are controlled by God or by fate
the doctrine that God has ordained all that will happen . . .
So I prefer inevitable — certain to happen. Some examples:
You must face the inevitable and try to deal with it.
Everybody is always trying to hide children from the horrors of life when it is inevitable that they will find them out in the near future.
When you're working such long hours, it's inevitable that your marriage will start to suffer.
Another relevant quotation rather than a single word is Henry Ford’s famous maxim which is generally quoted as “You can have any colour you want as long as it’s black”, but is actually “Any customer can have a car painted any color that he wants so long as it is black”.