Warlocks and Sorcerers share Charisma as their spellcasting ability and both can use an arcane focus. The Rod of the Pact Keeper is a focus for warlocks. Can a multiclass Warlock/Sorcerer use the Rod of the Pact Keeper as a focus for both spell casting classes but without the extra abilities the rod give the warlock?
Let's make a little checklist:
- Requires Attunement by a Warlock. You are a warlock, check.
- It's an arcane focus. Both of your classes can use an arcane focus, check.
There's no reason you wouldn't be able to use as an arcane focus for both classes since you have the only two requirements of the item. As you said, your sorcerer spells can't benefit from the bonus but the special ability only requires you to hold the rod, use an action and have a spent warlock spell slot.
Rod of the pact keeper isn't a spellcasting focus.
Here's the complete, current rules text for the rod of the pact keeper:
While holding this rod, you gain a bonus to spell attack rolls and to the saving throw DCs of your warlock spells. The bonus is determined by the rod’s rarity.
In addition, you can regain 1 warlock spell slot as an action while holding the rod. You can’t use this property again until you finish a long rest.
Nowhere does it say that the rod can be used as a spellcasting focus for any class.
A magic item must specify that it can be used as an arcane focus.
There are several magic items that belong1 to the list (orb, wand, crystal, staff, rod) of arcane focus types, and each of them specifies that it can be used as an arcane focus: see for example the Astral Shard (crystal), the Orb of Shielding (orb), the Ruby Weave Gem (crystal).
Since the Rod of the Pact Keeper does not list this option, it cannot be used in first instance as an arcane focus.
An item designed for being an arcane focus could be used as such by a multiclass spellcaster.
If an item, magic or not, is specifically designed to be an arcane focus then it could be used as focus by both casting features. For example, the Orb of Shielding states:
If you're a spellcaster, you can use this orb as a spellcasting focus.
A Warlock/Sorcerer can hence use this magic orb as an arcane focus for spells belonging to both classes.
There are other magic items that are arcane focus for a specific class. See the Dark Shard Amulet:
You can use the amulet as a spellcasting focus for your warlock spells.
In this case, such item works as an arcane focus only for Warlock spells, and cannot be used for Sorcerer ones.
1 See this answer of mine for other magic items that do not belong to the types list of arcane focuses but they can be employed as such.