i would split your water in your smaller tank.
put half the water from your older tank(if you are keeping it if not all of it) that way you get a lot of the natural bacterias needed then you would probably only have to give it a week cycling
Not exactly true. Water carries practically no bacteria (to colonise the filter) and will be of very little significant help - and most certainly will not have the other tank up and running fully established in a week.
But what will most certainly be of great help to colonise the new filter, would be to take some of the filter media from your existing filter and put that in your new filter. This method to "clone" tanks are very often used.
You will then need to monitor your ammonia and nitrite levels for a couple of weeks to make sure those remain at zero. And if there is a spike, to perform 20% daily water changes to help rectify this
Edit: the gravel and stocking method is usually also of very little benefit - again, as the vast majority of the beneficial bacteria resides in the filter media.
If you are upgrading your tank (and not keeping the smaller tank) simply add all the filter media from the small tank into the large tank. That should then be absolutely fine