I'm not "the little woman" (in any sense....) but I do the packing. It works for us.
DH works from home, does all the shopping, is available to take delivery of packages and deal with all tradesmen, makes all the arrangements for stuff like tradesmen and a huge proportion of the household paperwork, keeps us right with car servicing and MOTs, and used to do about 9/10 of the child-fetching till DS got older.
OTOH I commute to work, cook meals, do more of the gardening and cleaning, and put his clothes and stuff into a suitcase now and again when he (or we) are going on a trip. It's not that he's incapable of it, just that it gives us time together before he heads off for a few days - he decides what to bring and fetches it all together, and I fold it neatly (which I happen to be better at) and fit it ito the case. I reckon years of playing Tetris may have contributed to my ability to get it all in without sitting on the case....
I don't get why this one specific issue causes you such a problem? If men are incapable of household stuff, there will be a lot of bigger issues than packing a few times in a year. And if they are fine generally, and the relationship is an effective partnership, then how a couple arrange this one specific thing is no big deal, surely?