I programmed HaChu for a slight variation on Mighty-Lion Chess (without castling or e.p., and without the rule that protection by King for a Lion does not count), and had it play a few games in self play. These looked like reasonable games, not at all like a Lion-feeding feast. Now and then a Pawn got trampled by a Lion. So it seems not a major concern, even with the regular FIDE pieces. Worse, IMO, is the scarcity of trading opportunities for Lion against other material.

Castling was indeed intended to be the regular Capablance 3-step castling.

I did not know of the progressive promotion rule. It is an interesting option to encourage deferral. In Chu Shogi deferring Pawn promotion is pointles, as Gold is upward compatible with Pawn. For some other pieces it can make sense, but then they are not allowed to promote before they leave the zone again, unless they capture. (Yet they have a special rule that Pawns, which obviously cannot leave the zone, always promote on last Rank.) What I originally had in mind whas the same rule as I proposed for Mighty Lion: free choice but no Lion. The latter to avoid the possibility of having multiple Lions, with the resulting need to further complexify the rules on Lion trading. That argument still applies here.

The Grand-Chess rule of only allowing promotion to captured pieces would also achieve that, though. It would also provide an incentive for deferral and under-promotion. With the only hippogonal choice being the weak Knight, the latter would otherwise be just as uncommon as in Chess. For the game the promote-to-captured rule would probably be the best one; the only misgivings I have is that it modifies orthodox Chess in a way not towards Chu Shogi. A slight refinement could perhaps be the Great-Shatranj rule, that you always have the option to promote to the comparatively weak Commoner. In practice, this is a bit of a moot point, however. It is quite unthinkable that an opponent would allow any surviving promoted piece so early in the game.

Having the Commoner as fall-back option give me an idea for an unusual promotion rule that combines some of the rules you mention: Pawns can promote only (and always) to Commoner on the 8th Rank (and as deferring would be pointless, it is better to forbid it). This is sort of Makruk-like promotion. Commoners, however, can further promote to any captured piece when reaching 10th rank. This could also include the two primordial Commoners, and thus would introduce two concepts found in Chu Shogi: Pawns only promote to a not-so-strong piece, and pieces other than Pawn might also be able to promote.

To make it even more Chu-like, it might be better to eliminate promotion choice alltogether, and also have fixed promotion for the Commoner. This would then be Queen. In stead of promoting Commoners when they reach 10th rank they could promote when they enter the zone; both would be two moves away from the first (Pawn) promotion. On capturing in the zone they could also promote.

The concern is drawishness, however. In Chess Pawn endings, the first one to promote usually wins. Even if the other is just one step short, but and supported by King, you usually still can win (except for Rook and Bishop Pawns). But if the other would already have promoted to Commoner by the time you get your Queen, it becomes much more difficult to stop its promotion. And even KQKM itself would already be a drawn end-game when King and Commoner connect. On the other hand, due to the larger board it could be very unlikely that a Pawn is actually supported by the King, and when they are far apart it is comparatively easy to stop Commoner promotion and gobble it up with a checking fork. Perhaps this should also be tested in practice. Come to think of it, I am not so sure that KQKP can be won in Grand Chess when the Pawn is one step from promotion, due to the possibility for the King to protect the promotion square from deeper within the zone, when you stop attacking the Pawn. Another idle thought: even when you have the choice between Lion and Queen, the obviously weaker Queen will often be the better choice, because it can prevent the opponent from promoting his passer to a Lion. 

As to availability of pieces in over-the-board play, I don't think this should be a major concern in designing the rules. To play this game would require at least two Chess sets, even if just for the extra Pawns. Draughts chips could be used to 'decorate' the remaining pieces: Crowned Bishop and Rook would quite naturally be a Bishop and Rook placed on a pedestal. The Commoners could be represented by a Pawn on a pedestal, and a Lion (which is the ultimate super-Knight) by a Knight on a pedestal (perhaps a very high pedestal). This would leave all the usual associations Chess players have for the pieces fully valid, and extend them with a single universal new one: pedestal moves as King. (And it would also make it sort of logical that the 'Crowned Pawn' would be able to promote!) And it uses only half the pieces of the second Chess set, the other half (including Queen) remaining available for promotions.

[Edit]
This business with the pedestals actually does give me an idea. There could be two versions of this game, differing only in promotion rules: a more Chess-like and a more Shogi-like version. The latter would simply have every piece that moves into the promotion zone step on a pedestal. Rooks would become Crowned Rooks, Bishops Crowned Bishops, Pawns would become Commoners. Knights would become Lions! (But perhaps only if you don't have a Lion already.) They would have the role of the Kylin in Chu Shogi. Pawns (crowned or not) would no longer promote on last rank in the Chu version. Deferral would be pointless, except for the forced deferral when a Knight could not promote to Lion because you already have one. With the existence of this Shogi-like version, there would be no need to make the promotion rules of the Chess-like version more reminescent of Shogi. It could simply become free choice 8th rank, subject to the one-Lion rule. Which in practice means Lion (if available), Knight or Queen (the latter upward compatible with anything else). Commoners, being Pawns on a pedestal, would promote the same as other Pawns, so there would be no such thing as promotion to Commoner.