This DM would require the creator to research a new rod
Just to clarify: The creature that possesses the feat Craft Rod (Player's Handbook 92) researched how to create and then actually created a rod of shield, a DM-approved new magic item (Dungeon Master's Guide 198). When the DM approved that magic item, that magic item in its description had a caster level of 1. Now the creator of that rod of shield wants to increase the caster level of the shield spell effect that the rod of shield creates and add to that rod of shield's abilities the ability to create a dispel magic effect. (That's complicated enough that if any of that is wrong, the question should be updated!)
With this in mind, this DM would have the rod creator research how to create a new magic item, a rod of dispel magic. Then, if successful, instead of the creator actually creating a rod of dispel magic, the creator can add a rod of dispel magic to the creator's existing rod of shield by spending the time, gp, and XP necessary for creating a rod of dispel magic and adding to the price of the combined rod of shield and dispel magic half the price of less expensive item.
Essentially, like this fine answer says, a new ability is being added to the rod of shield, but that new ability is one that doesn't yet exist as a magic rod. The rod creator needs to puzzle out first how to make a rod of dispel magic—a magic item that has heretofore evaded construction by the mighty wizards who came before him—before he can add a dispel magic effect to an existing rod. Neither the presence of the feat Craft Rod nor the existence of Table 7–33: Estimating Magic Item Gold Piece Values (DMG 285) grants a creature carte blanche to add whimsically effects to a magic item!
"Should every new magic item be subject to the rigors of spell research as the rules say they should?"
No. Absolutely not. I mean, seriously, if a creature that possesses the feat Craft Wondrous Item (PH 92–3) wants a chaos diamond (DMG 252) (160,000 gp; 0 lbs.) as a half-price chaos hat, this DM's totally okay with that. It kind of ruins the whole magic gem vibe, but the game's not gonna implode. And, usually, this DM'd be on board with creating either a rod of shield or a rod of dispel magic (except that they're for some reason rods—see below) with an out-of-game handshake or whatever, rules be damned. However, not creating that rod of dispel magic—or, at least, not figuring out how to create it—and jumping straight to adding a brand new ability to the rod of shield for no reason and out of the blue? That, to this DM, goes a bit too far: in this DM's campaigns, the player and DM need to decide how the rod craftsman developed that new ability that otherwise for a rod doesn't exist. If it's through a method like spell research (as the game's rules detail), that's one way, and if it's through the DM telling the player, "Just have Harry Cobbler study the dispel magic spell in his spellbook for a day, and he can figure out how to add that dispel magic ability to his rod of shield," that's a perfectly legit other way.
Increasing a magic item's caster level after the item's creation
Improving an existing magic item's caster level is a trickier issue, but, despite the idea giving this reader a vague and undefinable sense of unease, this DM would allow it. That is, this DM doesn't see right now at this very moment any problems that could arise with upgrading an existing magic item's caster level after the magic item's creation. Magic items can already be changed—and it's implied that those changes can be to magic items' caster levels—after their creation, as demonstrated by this magic sword example from Adding New Abilities: "[A] +1 longsword [that has a prerequisite caster level to create of 3 but that has a caster level that the creator can set as low as 1] can be made into a +2 vorpal longsword [that has a prerequisite caster level to create of 15 but that has a caster level that the creator may be able to set as low as, like, 11], with the cost to create it [i.e 96,000 gp] being equal to that of a +2 vorpal sword minus the cost of a +1 sword" (DMG 288).
While that makes formalizing this process—therefore also allowing a creature that possesses the feat Create Wand to upgrade an existing wand's caster level, for instance—an exercise in extrapolating from the existing rules, and that is, to be sure, usually a risky proposition, in this case it's risky and interesting. Formalizing this extrapolation offers, for example, at least some small incentive for PCs to keep for later upgrading random found-during-an-adventure magic items that would otherwise be vendor trash. That seems like a good thing.
Why is this a magic rod?
I suspect the rod's creator may have missed what this reader has long thought the very point of magic rods: A magic rod is an excuse for a magic item to have an effect that can't normally be duplicated by spells. A magic rod is for facilitating acts of magic that are unusual, wacky, and magnificent… not for an effect so gauche and pedestrian as enabling the bearer to use 1/day an effect like the dispel magic spell much less the shield spell! Good sir, that rod—declares this reader on whose bookshelf sits a well-thumbed 4-volume set of the Encyclopedia Magica—has no business being a rod! Harrumph!
(Actually, this DM would've ruled that that a 1/day shield magic item be, instead, just a wondrous item. Now, if the shield effect protected up to 4 adjacent Medium creatures and looked like a Colossal Vin Diesel, then it's a magic rod!)