The 1ms sample interval limit is set by the PicoLog software. A relatively faster sampling rate can be set when using the Software Development Kit (SDK) in order to create your own application.
The PicoScope 3403D will only work with the PicoScope 6 software or with an application created using the SDK but not with the PicoLog software.
Collecting data at 4kS/s with 4 channels enabled for 4 hours would result in just under 220MS so it would be better to collect data in streaming mode (refer to our Library A - Z of PC Oscilloscopes page
for further information).
Having checked with the PicoScope 6 software, the maximum sampling rate with a 2000s/div timebase is 1.25kS/s with all 4 channels enabled on a PicoScope 3404D (this uses the same driver as the 3403D) so you would have to collect data over a shorter period of time, or use the SDK. There is a maximum record length of 100MS shared between all channels when collecting data in streaming mode using the PicoScope 6 software.
The software will stop data collection while it saves data to disk before restarting capture (assuming that this has been set in the Alarms dialog).
Hope this helps.