No, you can not run a BS report by selected dates. However, if you run a report as of March 31st, that is the same numbers for the beginning balances for April 1st. So if you select the March Cost Basis Balance Sheet and then go in and change the header to read April 1st, that would still be the correct numbers. Once report is on screen,
select Preview | Layout
select Header tab
Type in First day of month (highlight Month-end date and type over.)
Print.
Keep in mind, this Header change is only good for the one printing. If you go back to that report, it will revert back to the default setting of the previous month-end date.
As for the first month of your fiscal year, then of course you would have to use the last month of the previous year and change the header from that report.
If you keep your books at cost basis, then the Starting Balances report would be accurate. If you keep your books at FMV, then the Starting Balances report would not work for you.