First and foremost, you must have a current valid position in company Y to either enter on H1B or COS to H1B once you are in the US. You may not use entry on your L1 to "look" for a job; you already have an employer who is responsible for placing you in a position. Since you are not willing to enter using your H1B stamping, it appears that you do not have a valid employer-employee realtionship at this time with company Y.
You will be denied entry on H1B without current documents.
On a change of status, you may not begin working until you receive an approval notice and have in your posession a valid H1B I-94 either from entry on H1B stamping or accompanying the approval notice. You must be able to demonstrate you are in status when you file for the COS. You should have pay records from your home country as well as your per diem records.
As for issues changing from L1 to H1B in a valid situation, the USCIS will not have a problem but employer X may.

