I want the date to be parsed as is, and for no locale adjustments (in this instance, BST) to occur
That is exactly what should occur, however it doesn't in all browsers. You should not parse strings using the Date constructor or Date.parse (they are equivalent for parsing). Always manually parse strings, a library can help but usually isn't necessary.
According to EMCAScript 2015, '2016-06-03T09:05:15' should be parsed as a "local" date (i.e. based on the host system settings for date, time and time zone on the date and time supplied). A Date object's time value is UTC, so when creating the time value, the host settings are taken into consideration. The same settings are also used for output, so if the OP string is correctly parsed and then written to output, you should get back exactly the same date and time (though probably in a different format).
If you're seeing a different time from the input string, then the string isn't being correctly parsed (hence advice to manually parse strings).
Is this achievable without writing my own date/time parser?
Yes, use a library that someone else wrote. However, if you also want host settings to be ignored for output, you'll also need to write your own formatter, or use a library.
The good news is that there are many to choose from.