No, but...

If you duplicate a spell effect and the save DC that comes out is for a 7th-level spell, then it seems to me that the effect you're getting is a 7th-level effect. (And thus not suitable for contingency, ring of spell storing, etc.)

But Limited Wish also says it can "Produce any other effect whose power level is in line with the above effects". It seems like you could just wish:

• "I wish that my contingency spell should get a casting of reincarnate plugged into it."
• "I wish that my ring of spell storing should get a tongues spell plugged into it."
• "I wish to meet the 'imbue with spell-like ability spell' prerequisite while magic item crafting today."

I think all these effects are "in line" with the original power of duplicating spell effects.

There is no statuette so the requirement is always met

Wish removes the need for any requirements including material components for spells.

You don't need to meet any requirements in that spell, including costly Components.

Note that it doesn't say that wish gives you any spell components that you are missing, just that the requirements no longer apply. This is the same as spell restrictions based on class, for example. If the Bard wants to cast a warlock spell using wish they can do so and the spell doesn't magically make them a warlock for it to happen. The spell simply requires nothing for you to perform when using wish.

Contingency normally requires a statuette of yourself worth at least 1500 gp and ends if the statuette which was used as a material component for the spell is ever apart from you:

...contingency ends on you if its material component is ever not on your person.

However, because you cast with wish there was no material component (statuette). That requirement was completely waived by wish. Thus, the requirement simply does not apply. And thus, it essentially is treated as if the statuette was always on your person.