USB-C Splitter for Power-in & A/V-out - Does this exist?

I have a Lenovo ThinkPad laptop that has a single USB-C port (plus a couple of standard USB 3 ports and an HDMI port). The USB-C also happens to be the main charging port for the laptop.

I have a portable monitor, which connects perfectly to my laptop via a USB-C to USB-C cable, and is able to receive both power and A/V input via this connection. However, as the USB-C on my laptop is the only charging port, this means I can't charge my laptop and use the monitor at the same time using USB-C alone.

My solution has been to use a USB-C to USB-C/HDMI/USB-A splitter, in the configuration shown in the attached image (below). While this works fine, it's a bit "messy", as it requires two cables going to the monitor (HDMI [for A/V], USB-A [for power]) rather than a single USB-C cable, as is the case when the laptop doesn't require charging.

USB-C to USB-A/HDMI/USB-C splitter

I have been unable to find a USB-C splitter (male --> 2 x female) that would be suitable for this, however. Most appear to split to power and audio output, which means it wouldn't be suitable for POWER-IN (to charge the laptop) and A/V + POWER-OUT (for the monitor).

Does such a splitter exist? And, indeed, can such a splitter exist, or is this a limitation in the technology (i.e., in terms of power management)?

Thanks in advance!

Does such a splitter exist?

No.

can such a splitter exist

No, HDMI will always need an USB client device (as in your picture), and can never be done with a passive splitter.

A/V via USB-C uses so called "alternate modes", and is not the same as HDMI.

You cannot even split USB-C into a "power" USB-C part and a "non-power" USB-C part, because power management is quite complicated, and needs the USB data lines for negotiation.

is this a limitation in the technology

Well, it works differently from how you think it works. If you want to call this a "limitation" is up to you.