You can paste with the mouse by clicking the middle mouse button.
If you have a two button mouse, you can run xvesa setup, to set it up so you can paste by clicking both buttons at the same time.
However, if Xorg is installed, xvesa setup won't run.
There are ways to get around this, but new users may find them difficult.
I suggest:
Preferred option
Have Tiny Core set up so you can paste with all mice by clicking the two buttons at the same time.
If someone can think of a reason why that is not a good idea, here is an alternate option.
Include a script, which runs independent of xvesa, and modifies .xsession in the same way as xvesa, so it can be run easily by new users even if Xorg is installed.