What chemical could dissolve low-density polyethylene?
The safest solvent might be supercritical carbon dioxide, $\ce{CO2}$. Supercritical carbon dioxide is also used in "green" dry-cleaning. Use for dissolving polymers requires a pressure vessel for ~5 to 50 MPa, able to work at ~40 to 100°C. There are (paywalled) articles at ACS and NCBI, if you want to investigate further.
According to Agilent, 1,2,4-Trichlorobenzene (TCB) is effective.
Some other solvents mentioned at Research Gate include toluene and xylene.


N.B. The aromatic solvents all are toxic, some quite so, and are volatile and flammable. All of them require temperatures near the melting point of polyethylene, ~100°C, to dissolve much of the polymer, so a reflux or pressurized system is required.

Supercritical $\ce{CO2}$, while not very toxic, can cause asphyxiation above ~5%, and the container is, effectively, a bomb containing a warm or hot pressurized fluid.