This is general comment - not D5100 specific

I assume that the exposure is NOT locked for all of the sequence but can vary. As light level changes. This may be an option. You need to be able to allow exposure to change - ie an auto setting.

You may need to experiment with spot / centre / wide exposure settings. Depending on which of these you use the foreground composition may have an undesired effect on lighting. eg with wide exposure setting the foreground may not increase in brightness as rapidly as the sky near the sun does - so the average scene exposure will be progressively lower as the sun rises. Changing to spot or centre may make a significant difference. This may be the single greatest thing you can easily do to improve things. (But, it may not work :-) ). This plus exposure bracketing and/or HDR may help a lot.

You can also probably set eg auto-ISO so the camera backs off in ISO setting as the scene brightens.

Try taking some shots yourself and find out what settings give you the results you want at various light levels and then look for an auto exposure strategy that works.

IF you can set the camera to shoot exposure brackets in time lapse mode then set them as wide as possible and use what works best from the sequence.

If the exposure is locked for the whole sequence (seems unlikely) you are "in trouble". Some solutions that comes to mind then are

• External timer that takes timed sequential shots. Camera will autoexpose each time. This would also allow exposure autobracketing or HDR which otherwise may not be available in sequential mode.

• Getting desperate - A graded neutral density filter which is moved with time or light level. This would be an excellent solution functionally even with autoexposure and is not actually very hard to achieve if you have appropriate resources - but is not the sort of thing you'd usually need to turn to.