Can I perform inter-VLAN routing on a Layer 3 switch using a single trunk cable connected to an unmanaged Layer 2 switch?

Can I perform inter-VLAN routing on a Layer 3 switch using a single trunk cable ...

Yes.

... connected to an unmanaged Layer 2 switch?

No. An unmanaged switch doesn't know about VLANs. Port-based VLANs are impossible and tagged VLANs are not limited by membership - any device connected to the switch can join and leave tagged VLANs as it pleases.

Practically, running tagged VLANs across an unmanaged switch does work most often, but there is no security added by doing so, and each end node needs to use tagging as desired. Also, there is only a single broadcast domain.

I'd like to avoid the manual work of tracing which endpoint connects to which port.

On a managed switch, just check the MAC/VLAN table. On an unmanaged switch, each end node controls its VLAN connectivity itself. I fail to see the point for VLANs in that setup.

Is there a way to do this without rewiring everything?

Use a managed switch and set up VLANs on it. There's no need to rewire anything.

Run multiple cables from the L3 switch to the unmanaged switch - basically one cable per VLAN

That won't work since the unmanaged switch just bridges all VLANs together.

The only alternative to a managed switch is using one unmanaged switch per VLAN and uplink each to an access port (a single untagged VLAN) on the L3 switch.

To your added diagram: of course that setup works in that both IP subnets can talk to each other. The point is that the VLANs are entirely useless since you bridge them together on the unmanaged switch. You'd get the exact same result with a single VLAN and binding of both IP addresses to that on the L3 switch. The problem is that any end node can change its IP address to the other subnet and simply use that. Or use both at the same time, bypassing the L3 switch and any ACLs you might have set up there. Any security added by VLAN separation is just an illusion.

In a nutshell: you need a managed access switch for VLANs to make sense here, or alternatively two unmanaged switches, one for each VLAN.

I suppose it's plausible that an unmanaged switch would forward frames based on just the MAC addresses, without regard for the VLAN tags any more than switches usually care about Ethertypes.

That is, a broadcast frame tagged for VLAN A would be flooded to every port with a tag for VLAN A. Similarly for unknown unicast. Regular unicast traffic should still work normally. The switch should keep each VLAN "intact", so to say, and not mix frames from VLAN A to VLAN B, but of course it wouldn't provide any control for which VLAN goes where, and any VLAN pruning would need to be done downstream.

In effect, it should be quite similar to taking a managed VLAN-aware switch and configuring all VLANs to all ports(*). Doable, for a sort of a hub switch where all connected devices are other switches under your control, so they can do VLAN pruning, and put each end-user device in the VLAN it belongs in. Not the prettiest thing to do since the downstream switches would receive stray frames from VLANs they won't need, but at least you have one place less to configure...

(* Well except that a proper VLAN aware switch should give separate MAC tables for each VLAN, which matters in case some particular MAC address appears in different ports in different VLANs, but that should be rare.)

Then again, it's also plausible that a dumb switch would just drop VLAN-tagged frames. (Perhaps if the switch chip is actually VLAN-aware, just configured with a static single-VLAN all-untagged configuration.) But even a dumb switch would seem quite unlikely to go swapping VLAN tags for other VLAN tags. Not that you can be sure without testing, and you likely won't have any guarantees.

But, in your image, you have what seems to be workstations, with labels like "admin" and "guest", no less. Here, using an all-VLANs-open configuration (with a managed or an unmanaged switch) would be an extremely bad idea, as there wouldn't be anything to prevent the "guest" workstation from sending and receiving frames on the "admin" VLAN.

If this was a homelab where the workstations are also only used by you, you could play with this, but NE.SE is supposed to be for "professionally managed networks" so just go get proper hardware. Switches aren't that expensive, and figuring out what goes and needs to go where in the network is pretty much part of the job description anyway.