In theory, you can be a shareholder, a director, and an employee. Many people setup pte ltd's in this manner.
Shareholders own the company. Your remuneration for sinking money into the company is the dividends (if any) that you receive. You don't need to have any kind of legal residential status to be a shareholder in a Singapore company.
Employees perform services for the company for remuneration, either for piece work, an hourly rate, or a periodic salary. Employees must have legal work authorization... citizen, PR, EP, SPass, or a DP and LOC. Directors who perform management functions, for example, hiring, signing contracts, prospecting for sales, etc, would receive a salary (not directors fees) for the work they do for the company.
Clearly, then, you can be an employee on a DP/LOC, so long as your existing director makes application for same.
Directors, in addition to being paid a salary for work performed, can also receive directors fees for performing directors duties, such as AGM's, EGM's, board meetings, etc. Directors in this capacity have full fiduciary responsibility for the company, and are often compensated with fees for taking on this risk.
Now, here's the rub... if you are a citizen or a PR, you can be any and all of these things. However (and I have been looking for the exact citation but cannot find), MoM has decided you cannot be a director on a DP/LOC. The reasoning is that you are not an employee, but you are firm management. Therefore, you must have an EP. Again, I can't find the reference but I know I have seen it somewhere.
Since you already have a normally resident Singapore director, the only reason that you would want to become a director would be to _share_ director's powers with your existing director, or, to replace that director with yourself.
You could try a couple of strategies. You could have your director set you up under a DP/LOC, then set yourself up as director in ACRA, and see what happens. You might not be able to do it if they have modified BizFile to check residency status.
Or, you could have your director apply for an EP for you as an employee of the company. You need to meet EP salary and educational requirements. If that application is successful, then you can readily be added to ACRA as a director. And because you have EP (i'm assuming you have a Singapore address), you are considered normally resident, and therefore can be the sole director.