We get different values each time because the fit is generated from random data, but I think you can get the expression as follows, and can use it to define a new function as needed. Yes?

In[23]:= exp = Normal[fit]

Out[23]= -20.981 E^(-50.6429 (-3.00695 + t)^2) + 
 18.0241 E^(-0.257059 (-2.65573 + t)^2)

In[24]:= term1 = exp[[1]]

Out[24]= -20.981 E^(-50.6429 (-3.00695 + t)^2)

In[25]:= t1[t_] = term1

Out[25]= -20.981 E^(-50.6429 (-3.00695 + t)^2)

The Normal[fit] consists of two summands: -137.812 E^(-18.4523 (-11.7337 + t)^2) and 15.2716 E^(-0.525651 (-1.92061 + t)^2)

I need the first of them: Normal[fit][[1]], this outputs the first part, but I can't plot it or do numerical computations on it as I posted above.

This will use the fitted model to define a function, but it is not clear there is an advantage over using fit[t], as you did in your plot.

In[5]:= f[t_] = Normal[fit]

Out[5]= -137.812 E^(-18.4523 (-11.7337 + t)^2) + 
 15.2716 E^(-0.525651 (-1.92061 + t)^2)