I'm a month late on reply but at least you have an answer from a customer using LiteSpeed...
I'm using LiteSpeed on a VPS. I did transfer and restore 22 accounts that were not using LiteSpeed before and no issue at all after the transfer. You can even switch(a single click!) between Apache and LiteSpeed to see for yourself the difference in both software.
I didn't face any PHP issue after the transfer. I did not have any account with php.ini files but all .htaccess did work right away.
If you want, you can send me a php.ini which you think would not work and I'll try them to see if LiteSpeed throws errors.