It’s fine to upgrade DifferentiationInterface. I just put the two together and checked:
(@v1.10) pkg> up
Updating registry at `C:\Users\accou\.julia\registries\General.toml`
Installed BoundaryValueDiffEqFIRK ───── v1.5.0
Installed NLSolversBase ─────────────── v7.8.3
Installed VertexSafeGraphs ──────────── v0.2.0
Installed BoundaryValueDiffEqShooting ─ v1.5.0
Installed BoundaryValueDiffEqMIRK ───── v1.5.0
Installed PackageExtensionCompat ────── v1.0.2
Installed SparseDiffTools ───────────── v2.26.0
Installed Optim ─────────────────────── v1.11.0
Installed DifferentiationInterface ──── v0.7.0
Updating `C:\Users\accou\.julia\environments\v1.10\Project.toml`
[a0c0ee7d] ↑ DifferentiationInterface v0.6.54 ⇒ v0.7.0
Updating `C:\Users\accou\.julia\environments\v1.10\Manifest.toml`
⌃ [85d9eb09] ↓ BoundaryValueDiffEqFIRK v1.7.0 ⇒ v1.5.0
⌃ [1a22d4ce] ↓ BoundaryValueDiffEqMIRK v1.7.0 ⇒ v1.5.0
⌃ [ed55bfe0] ↓ BoundaryValueDiffEqShooting v1.7.0 ⇒ v1.5.0
[a0c0ee7d] ↑ DifferentiationInterface v0.6.54 ⇒ v0.7.0
⌅ [f6369f11] ↓ ForwardDiff v1.0.1 ⇒ v0.10.38
⌃ [d41bc354] ↓ NLSolversBase v7.9.1 ⇒ v7.8.3
⌃ [429524aa] ↓ Optim v1.12.0 ⇒ v1.11.0
[65ce6f38] + PackageExtensionCompat v1.0.2
⌅ [c0aeaf25] ↓ SciMLOperators v1.3.1 ⇒ v0.4.0
[47a9eef4] + SparseDiffTools v2.26.0
[19fa3120] + VertexSafeGraphs v0.2.0
There is nothing else in your Project.toml? Just those two are fine.