Why do you think white 'gets too much'? In normal Chess there are many acceptable replies to white's first move. So white's first placement of the Duck is just as inconsequential as an extra first placement by black at the start of the game. Do you really have any evidence that the white advantage in Duck Chess is any larger than with FIDE rules? You might be trying to solve a non-existing problem, sacrificing simplicity for no benefit at all.
How to improve upon Duck Chess
I made a simple demo applet for Duck Chess. It doesn't play perfectly, but it should be fun to beat it:
https://www.chessvariants.com/rules/duckchess

Why do you think white 'gets too much'? In normal Chess there are many acceptable replies to white's first move. So white's first placement of the Duck is just as inconsequential as an extra first placement by black at the start of the game. Do you really have any evidence that the white advantage in Duck Chess is any larger than with FIDE rules? You might be trying to solve a non-existing problem, sacrificing simplicity for no benefit at all.
Ummm, you might want to re-read my original post. It's like you're not only arguing with the person who suggested that White got too much (which is not me), but also offering the same rebuttal as I did to them. 😆
I'll also make sure to check your app, but in the meantime, let's return to the subject. Let's say that we agree - and we do - that positioning the duck on the first move (or even before it) is a non-issue and doesn't affect the balance all that much. If so, what about the follow-ups, then?
Last time I played, I had the Black pieces and the game started like this: 1. e4-De6 Nc6-Dd3, after which my opponent and I reached some kind of a Four Knights Glek for both sides, and I felt that the position got very closed due to our mutual duck-blocking of the d3- and d6- squares. After who knows how many moves, I barely managed to get a Knight for a Bishop (which is almost certainly advantageous in this variant) and then went on to lose on time xD, but the real issue that I've found is that it's hard to open up the centre if both sides are insistent on the mutual restriction of Bishops and/or castling.
The play becomes forcibly closed and considerably slowed down, and there's no real way around it. That's why my suggestion is to push the initial duck positioning one move to the future and give it to Black, because I feel that it allows positions to be more vairied that way. Managing Bishops becomes a fair risk-reward possibility instead of a chore. What do you think?
That's the neat thing; you can't.
...
Well, actually, I've seen some common-sense complaints about duck chess, as well as some decent ideas on how to improve upon it. Here's how it evolved:
1) the first complaint I saw was that White gets too much by playing first AND placing the duck first, so the proposition was to make Black position the duck before White's first move, and that really does seem very reasonable;
2) the next complaint (by a different person) was that the above suggestion wouldn't make a significant difference, as White would still be able to fully move one of the two centre pawns without any drawbacks, and Black would stand to gain very little from self-blocking a move in order to free it up (because White can see it and play around it)... therefore, the next proposition was to allow any first move for both sides and place the duck only afterwards;
3) however, what I personally noticed as the drawback of the above implementation was the fact that White would get to develop one Bishop to a non-long diagonal completely freely and then just keep stopping Black from doing the same in return, which would be a greater imbalance than what we presently have as the official duck chess format... therefore, I proposed that the duck should be placed on Black's 2nd move, as White's first-move advantage would be perfectly offset by that, only without the hassle of the game devolving into Four Knights variations with fianchettos and perpetual duckblocks on squares e3, d3, e6 and d6.
I also thought whether it should be Black's 3rd move instead, but it's like you can develop both of your Bishops by move 3, so it would give too much freedom to some transpositions. I'd prefer if the players had to risk developing the Bishops early, knowing that they might not get another chance for a while.
Anyway, what do you think?