It is impossible to answer your question without knowing more about the position offered, its requirements, your background, the reasons for the L-1 denial, etc. If the position qualifies as a specialty occupation and you meet the degree requirements, you should be able to get a H-1B unless there was something in your application that makes you inadmissible to the US. Just because you had a prior visa denied does not necessarily cause problems in the future. However, the requirements for an L-1B blanket and for a H-1B are similar, so if the L-1 was denied because the position was not professional, a H-1B might have the same problems.
The benefit of a H-1B is that CIS adjudicates the petition, not the consulate, so the attorney may be able to address any concerns that CIS has before they decide.