1D 9-15 4+ diamonds unbalanced, or 13-15 balanced (2+)
Without a 4cM or a hand appropriate for an inverted minor raise, we bid balanced hands as follows:
1D-1N 6-10
1D-2N 11-13
1D-3N 14+
However, our wide ranging openings are giving us some trouble. In particular, after 1D-1N, opener with a balanced 14-15 count might be tempted to invite game (with 2N) in case responder had 9-10 points for his 1N response. Of course this will work out poorly if responder only had 6-7. Similarly, after 1D-2N we might be too high without a fit if opener only has ~9-11 points and a semibalanced hand. Even when responder has 14, he might not make 3N opposite opener's dead minimum.
What do you think about this?
1D-1N 5-9, no game interest (avoids tempting opener to stretch for 3N)
1D-2N 13-14, a little sounder invitation is safer in case opener is minimal
1D-3N 15+, if you can't game force with a 15 count you're opening too light!

which leaves my suggestion of adding the extra 10-12 range to the otherwise natural 1M response:
1D-1M could be better major with 10-12 balanced, maybe only 3 cards
I've heard Meckwell bid 1M not infrequently on only 3 cards and that for a while in standard some problem hands were handled this way so this is not without precedent. If opener has the reasonably frequent 13-15 balanced hand and the auction starts 1D-1M-1N, we have good methods (2-way checkback) and handling the 10-12 balanced hand by bidding or invite game accurately should be easy.
The question is then whether we can handle the subsequent auctions if opener makes an inconvenient raise of our 3 card major after the 1D-1M. Ideally we'd like to avoid those unfortunate 3-3 fits when opener raises on 3 cards with a ruffing value, for example, although we also don't want opener insisting on game in the major (in the 4-3) just because he has 4 card support for our "suit". Of course since partner is limited to 15 points, he can't have a huge hand which helps somewhat. Typically our responses are:
1D-1M-2M some 3 card raises or a minimum-to-intermediate 4 card raise
1D-1M-3M strong 4 card raise
1D-1M-4M rare (and we hope partner won't bid this without 5 cards)
After a 2M raise, 2N is normally a natural invitation with 4 cards (catering to opener's 3 card raise). We might have to adjust this by playing artificial methods, something many advanced partnerships do already on this sequence. After a 3M raise, we'd need to use 3N as a more strong preference for NT than merely "choice of games". Probably with some rearrangements we could allow opener's normal "higher than 3N" bids (splinters, etc) into some sort of lower relay (jump reverse maybe?) that gives responder an out in 3N. All this is by way of saying it seems possible if a little complicated to restructure opener's typical rebids to avoid hanging us when we have only a 3 card major.
What do you think?