Is it possible to do linear interpolation on a manual E6-b?
My advice in this case would probably be: Once you know the density altitude, don't bother with the flight computer. The exact values are going to depend enough on the specific airplane that a general-purpose device will only get you approximations anyway, no matter what you do.

Rather, consider that for your fictious case, when going from a density altitude of 0 feet to 2,000 feet, takeoff distance increases from 1,865 feet to 2,602 feet.

Split the difference into reasonable, approximately equal-sized chunks. Say density altitude 0 ft, 700 ft, 1,400 ft, 2,000 ft. Calculate take-off distance for those, possibly based on a linear extrapolation (which isn't going to be exact), and note the change in take-off distance per change in density altitude. In your example case, for every 700 ft of density altitude increase in this range, the take-off distance increases by about 250 ft. Alternatively, for every 1,000 ft of density altitude increase in this range, take-off distance increases by about 370 ft. (If you're in a metric country, this happens to work out nicely to a little over 100 meters of runway per 1,000 ft of density altitude increase, so for a first order approximation you could just cross over the final digit in the density altitude in feet.) Note these values.

Now suppose that the density altitude for your airport of choice on the day and time of your flight is 1,370 ft. That's reasonably close to 2,000 ft minus 700 ft, so take the take-off distance for 2,000 ft (2,600 ft, give or take) and subtract 250 ft to get an approximate value of 2,350 ft take-off distance for this density altitude.

Or suppose the density altitude is 491 ft. That's closer to 700 ft than it is to 0 ft, so take the take-off distance for 0 ft (1,865 ft) and add 250 ft. That gives about 2,100 ft take-off distance. If you split into thousands, you could take half of 370 ft or about 200 ft, for a hair over 2,050 ft take-off distance.

For your example case, if the last hundred-ish or so feet of runway are going to be what makes or breaks the take-off, then I suspect you're cutting it too close anyway, so this level of accuracy should be sufficient for general aviation. With this, I'd probably have a rule of my own saying that if the runway length is less than, say, 150% of the calculated take-off distance, it needs a more accurate calculation or manufacturer performance data lookup; if the runway is less than 125% of the take-off distance thus obtained, it's a no-go. Remember that you have to be able to safely abort the takeoff even late during the takeoff roll!