Sony Handycam DCR-PC1E

I'm not from the UK so can't help you find a lab.
but regarding your problem -

The fact that it works well without a tape tells us that the load motor and possibly even the capstan motor and associated mechanisms are working. It doesn't tell us much about anything else though.

Could be that the head drum is not spinning - This can be the result of a siezed drum motor, or power supply fault(if seized it will draw excess current, thus giving a battery warning indication).

A faulty, or possibly even a correcly working dew sensor - This will allow the heads to spin, but prevent the tape from actually lacing up.
This is a 'hard' one really, because if the dew sensor is faulty you will get this problem, but you will get the same problem when the sensor is working and actually doing its job in that it has really detected moisture.

There are many other possible causes too.

The first thing to look for is to see if the heads are spinning - or at least see if they *try* to spin, because unless they *are* spinning the tape will never load.
Also, the heads *shouldn't* be spinning with no tape inserted - if they are then this suggests that the dew sensor has been activated (the heads are made to spin under this condition in an attempt to dry things out).

If this gives no clues, the next step would probably be to dismantle the thing and attempt to load a tape by hand (looking for signs of jamming mechanism, etc.

Anything more serious than this is one for the workshop.

Platano guatemalteco?

Good luck!