At the risk of revealing that I have completely misunderstood your question, a few thoughts...

People sometimes talk about regular QM as being like "zero-dimensional QFT,"* and I think that correspondence is more or less what you are getting at here. I'm not sure to what extent this viewpoint has been or can be formalized. But here is my understanding of the intuitive content.

What I will call the "quantum mechanic limit of QFT"** is like if you shrunk your string to a point, so that the amplitude is finally fixed at some trivial value (if you like, the two boundries join into one). But, this limit is taken in such a way that the mode structure remains intact.

Because the amplitude is fixed, both the spatial mode you call $\phi_n(x)$ and what you call the wave function become trivial. All that is left is the occupation number, which of course can be expanded in various bases including the energy eigenbasis (which makes it looks closer to QFT) or the position basis (which looks rather different).

A formulation of QFT that is helpful in this context is the wave-functional approach (see for example this question), in which the fundamental object is $\Psi[\phi(x)]$, a probability amplitude distribution over field configurations. In this notation, the QM limit would be one in which the field configurations $\phi$ all become trivial, but there is still some probability amplitude distribution over them.

*Specifically, I have heard this kind of language in the context of statistical physics. One can map a classical field theory in $d$ dimensions to a quantum theory in $d-1$ plus imaginary time. When $d=1$, the resulting quantum theory looks like a single-particle Hamiltonian.

**To be clear, this is completely different than a non-relativistic or particle conserving limit of QFT, which one might also call in a different context the "QM limit"

At the risk of revealing that I have completely misunderstood your question, a few thoughts...

People sometimes talk about regular QM as being like "zero-dimensional QFT,"* and I think that correspondence is more or less what you are getting at here. I'm not sure to what extent this viewpoint has been or can be formalized. But here is my understanding of the intuitive content.

What I will call the "quantum mechanic limit of QFT"** is like if you shrunk your string to a point, so that the amplitude is finally fixed at some trivial value (if you like, the two boundries join into one). But, this limit is taken in such a way that the mode structure remains intact.

Because the amplitude is fixed, both the spatial mode you call $\phi_n(x)$ and what you call the wave function become trivial. All that is left is the occupation number, which of course can be expanded in various bases including the energy eigenbasis (which makes it looks closer to QFT) or the position basis (which looks rather different).

A formulation of QFT that is helpful in this context is the wave-functional approach (see for example this question), in which the fundamental object is $\Psi[\phi(x)]$, a probability amplitude distribution over field configurations. In this notation, the QM limit would be one in which the field configurations $\phi$ all become trivial, but there is still some probability amplitude distribution over them.

*Specifically, I have heard this kind of language in the context of statistical physics. One can map a classical field theory in $d$ dimensions to a quantum theory in $d-1$ plus imaginary time. When $d=1$, the resulting quantum theory looks like a single-particle Hamiltonian.

**To be clear, this is completely different than a non-relativistic or particle conserving limit of QFT, which one might also call in a different context the "QM limit"