How to find peak value of an analog signal efficiently after sampling in the digital domain?
You are unlikely to be able to evaluate correctly an ideal sinc interpolation filter in real-time (or even with a small time lag), since the future samples of your data are unknown. Plus, when you received your last sample $x[n]$, you generally don't know whether you are on  a peak.  So, it somehow boils down to

how real-time you want to be;
how fast you can estimated you are passed a peak.

There are many local possible interpolations. You can also think about  fast option, and close to real-time, only require to collect the next sample $x[n+1]$, and if knowing $x[n-1]$ you consider that a peak is located between  $x[n-1]$ and $x[n+1]$ (and there values are positive), you can use the standard frequency peak  estimators, which only use the above three samples. For instance:
$$n_{\textrm{peak}} = n+\delta$$
with
$$\delta = \frac{(x_{n+1}−x_{n-1})}{(4x_{n} − 2x_{n-1} − 2x_{n+1})}$$
A primer is provided by famous people here: Fast, Accurate Frequency Estimators, Eric Jacobsen and
Peter Kootsookos, IEEE signal processing magazine, May 2007. The above equation is only one of those this paper provides.