If you are struggling with something, part of your job is to ask for more resources. If customer or stakeholder management are problem areas for you, then asking the organization for dedicated resources to manage those things for your project is perfectly reasonable. Whether or not those resources are allocated will determine whether or not you can remain effective on the project.
Analysis and Recommendations
Do I need strong account management skills to excel as a PM?
Not necessarily, but they do help. However, you probably need them if you plan to stay at your current job, especially if leadership can't or won't assign the necessary support resources to cover any gaps in your core capabilities.
While skills can be learned, personality and aptitude are largely ingrained. If your current job requires that you do things that you're really bad at, and don't see yourself improving in those areas, then you should look for a role that doesn't require you to do those things.
Project management is no different than any other career. You can find jobs with broader scope, narrower scope, or different scope. Since you seem pretty clear about the fact that you want to manage project internals rather than stakeholder or customer management, then look for a role where that's an option.
As some generic advice, smaller companies generally require people to take on more scope and wear more hats. Larger companies, especially those with a formal PMO, often have roles that are more clearly defined, and where you can get support from elsewhere in the organization to manage the things that aren't within your core capabilities.