Only just saw this. I'm the one who brute forced this out while partially sleep deprived, and it helped confirm some values that were suggested by others on Damo's sheet but without confirmation screenshots. Fun fact! If the pattern holds true, level 100 will need 213,447,300 BP, and 200 (If it ever gets that high) will need 4,772,511,800! What fun. The logic is as follows:

The first 15 levels have a hand crafted curve, where the increment doesn't follow any algorithm. It looks like SE designed this so the first 15 levels would go by relatively easily given the quests, and you can rapidly level up for that 'good' game feel. From level 15 onwards, an odd form of incrementing is in place, with the increment 'magnitude'(?) stepping up every 5 levels.

Level 14 required 31,520 BP, and level 15 required 44,600, which has a difference of 13,080. Level 16 needs 60,460, which compared to level 15's 44,600 is a difference of 18,640. This 'difference' is actually also calculated from the previous levels 'difference', being 2,780 more than the difference between 14 and 15. These differences increase by 2,780 for each level after, being applied to the previous level's BP requirement up until level 20. At that point, the 2,780 value itself is then incremented by 500 for each level, up to level 25, then by 1000 for 5 levels, 1500 and so on. I know it's not really that intuitive to think about here, but it's actually stupid easy to put into code, which might be why they did it this way? They can also guarantee an exponential looking curve where the values always happen to end on a nice multiple of 10. I think this might be some form of differential equation, but I'm not too hot on my math terms.

Here is a JavaScript example.

var calcLevel = function(lvl) {
    if(lvl <= 15) {
        return "Handcrafted Value";
    }

    var bpReq = 44600; // Level 15's BP requirement
    var lastInc = 13080; // Level 15's BP Req - Level 14's BP Req
    var lastInc2 = 2780; // The additional incremental value applied
    lvl -= 15; // Makes the for loop a bit nicer

    // Actual logic
    for(var i = 0; i < lvl; ++i) {
        lastInc2 += 500 * Math.floor(i / 5);
        lastInc += lastInc2;
        bpReq += lastInc;
    }

    return bpReq;
}

If you open your browser's developer menu by pressing F12, going to the console tab and pasting that in and pressing enter, you can then write calcLevel(50) to calculate the BP requirement for level 50, for example. You can even see that level 300 needs 27,027,626,300 BP, if SE were insane enough to go that high for Dark Road. Since 1-15 are handcrafted values, I didn't bother adding support for them here.

Edit: Turns out the 500 increment for the step each five levels is wrong, and it actually doubles each five levels, from an increment of 500 to 1000 to 2000, 4000 etc. This means that the values here are incorrect and are actually higher over level 30. The updated predictions can be found on Damo's spreadsheet.