If you don't mind knowing who your opponents are before the game starts, but still want to do it randomly, you could always do a manual random selection.
Basically run an online random number generator between 1 and 43. Select your civ, and then select each civ based on what the random number generator says, and if the RNG selects the civ you don't want to include, then don't include it and run it again.
It takes a bit of time, but considering it's a game of civ, something taking a lot of time is fine. Not wanting to know the civs beforehand is a bigger problem though. If you don't want to know the civs ahead of time and have access to a friend/family member nearby, you could also possibly do the same thing I said above, write down some numbers, and have the other person select the nth civ on the list (where n is each number you wrote down), while telling them to not select whatever civ you don't want. Not necessarily practical, but it's another way to simulate this effect without letting yourself know who you're up against. You just have to hope the person you ask doesn't hate you and put 7 Alexanders in the game for laughs.
An advanced option would be nice, but without mods, there are ways you can get around this if you're creative

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