Yes – after one restricts the variation of the metric to a fixed Kähler class, the Laplacian on functions still has a simple spectrum for a “generic’’ metric. In other words

For every compact complex manifold (M , J) and every Kähler class [ω0], the set
G = { ϕ ∈ C∞(M)/ ℝ : ωϕ := ω0 + i∂∂̄ϕ is Kähler
and all non–zero eigenvalues of Δϕ are simple } is residual (a countable intersection of open dense sets) in the space of C∞ Kähler potentials; equivalently its complement is meagre.

The statement was proved – essentially by repeating Uhlenbeck’s argument with only cosmetic changes – by

H. Urakawa, “Generic properties of eigenvalues of the Laplacian for Kähler metrics’’, Proc. Japan Acad. Ser. A 67 (1991), 350–353,

but the proof is short, so let us outline it.

1. The space of metrics one is allowed to vary in.
   Fix k ≥ 2n+4 and consider the Banach manifold
   Kk := { ωϕ = ω0 + i∂∂̄ϕ ; ϕ ∈ Ck+2(M, ℝ) , ωϕ > 0 } . Its tangent space at ωϕ consists of real (1,1)–forms of the form
   h = i∂∂̄ψ, ψ ∈ Ck+2(M, ℝ).

2. The universal eigenvalue equation.
   Following Uhlenbeck, consider

   F(ω , u , λ) = Δω u − λu        (u real–valued),          (1)
   

   together with the normalising conditions ∫Mu² dμω = 1 and ∫Mu dμω = 0. The map
   F : Kk × Hk(M,ℝ) × ℝ → Hk−2(M,ℝ) is C¹; an element (ω , u , λ) with F = 0 is an eigenpair for Δω.

3. Linearisation in the (u , λ)–direction.
   At a zero of F the derivative with respect to (u , λ)

      D(u,λ)F : (v , μ) ↦ Δω v − λ v − μ u                    (2)
   

   is a Fredholm operator of index 0. It is surjective exactly when λ is a simple eigenvalue.

4. Looking for regular metrics.
   We would like to show that for “most’’ ω the map (2) is surjective.
   By Sard–Smale it suffices to check that for every multiple eigenvalue there is a variation h = i∂∂̄ψ making (F, u, λ) transversal.

5. Producing enough variations inside the Kähler class.
   Write the first variation of the Laplacian in the direction h :

   (DωF)(h) = − ⟨h , Hessω u⟩ω   + lower–order terms.        (3)
   

   Pick a point p where u has a non–degenerate critical point. Choose local holomorphic coordinates in which Hessω u(p) is any prescribed real J–invariant quadratic form Q. Taking ψ(z) = ½Qαβ zα z̄β ·η(|z|) (η a bump function) one gets a variation h = i∂∂̄ψ that coincides with Q at p and vanishes outside a small ball. By (3)

   ∫M u (DωF)(h) dμω = − |∇²u(p)|²_{ω}  < 0 ,                 (4)
   

   so (DωF,h) is not in the range of (2). Hence (u , λ) is regular with respect to that h.

   Because one can repeat the construction for each pair of independent eigenfunctions in the same eigenspace, the degeneracy is removed.

6. Application of Sard–Smale.
   The set of regular values of the projection
   π : {(ω , u , λ) : F(ω , u , λ)=0} → Kk , π(ω , u , λ)=ω is residual in Kk; for any such regular ω every eigenvalue of Δω is simple. Passing to the Fréchet intersection k → ∞ gives the announced result for smooth metrics.

7. Remarks and variants.

   a) The zero eigenvalue is forced to have multiplicity one (constants).

   b) The argument shows that the generic Kähler metric has no continuous isometries. If the complex manifold carries a non-trivial holomorphic vector field, the metric breaking all its flows is still obtained by a small perturbation inside the Kähler class.

   c) The same method works for the Hodge Laplacian on (p,0)– and (0,p)–forms. In higher degrees additional symmetries (Lefschetz operators) create inevitable multiplicities.

Thus there is no “cunning counter-example’’: within any fixed Kähler class the metrics for which the Laplacian on functions has multiple eigenvalues form a meagre subset, exactly as in Uhlenbeck’s original unrestricted statement.