My answer/comments here (way too long for a SE comment ) don't and are not meant to respond to the particular question BUT as BACON once opined "To ask the proper question is half of knowing" . I am suggesting that the OP didn't ask what I think is the larger/proper question i.e. how to simulate/forecast future day's activity but rather focused on the issue of continuous vs discrete , a problem/opportunity in itself. Please don't downvote/criticize my response as not being pertinent to the question but rather being oriented to a bigger issue.

If you have 303 days of  data and you want to predict/simulate tomorrow , why would you use the observed history for all 303. The answer is ..iF you assume that tomorrow is like every other day then you are good to go. What we find is that if you actually model arrivals as it relates to day-of-the-week , you can then get a conditional distribution for tomorrow reflecting the day-of-the-week using the residuals from a simple (perhaps toooo simple !) model. The residuals reflect the conditional distribution around the expected value for tomorrow and can be harvested to provide a monte-carlo distribution for tomorrow. 


Now just a little bit more realistic. If there are unusual values (the source of your fat-tailed distribution ) they can be identified along with ant time trends, level shifts , day-of-the-month effects and of course memory effects to effectively construct a conditional distribution reflecting unknown/un-identified sources of variation. With a "richer model" we will be able to get a better expectation for tomorrow and the uncertainty in that expectation.

Now the good news is that if anomalies ( one-time pulses) have been detected and remedied to make a prediction ( N.B. all predictions are simulations and all simulations are predictions ..they are synonyms ) it is now possible (and correct) to enable the possibility of anomalies occurring in the next period.
If the next day is a Monday and no anomalies have been observed in the past on any Monday then pulses will b expected tomorrow BUT if previous Mondays have been effected then anomalies will be appropriately pro-rated.

I point you to to a reference that discusses the simulation/prediction of activity for the demand for daily cash  cash.http://autobox.com/cms/index.php/afs-university/intro-to-forecasting/doc_download/53-capabilities-presentation slide 49 ..

In summary the statistical action is all about the residuals as they are equivalent to an adjusted observation incorporating factors that can reflect/explain identified variation e.g. holiday effects and even particular day-of-the-month effects


Incorporation of forecasting methods as a precursor to "simulation" is  clearly on the horizon (so to speak !).