To address the remaining issue:
I've never seen it written anywhere that you have to use a specific implement for implement powers
"Implement: If the class can use one or more kinds of implements, that information is noted here."
PHB page 52
Note that the default assumption is that you can't use any kind of implements.
"A warlock wielding a magic rod or wand can add its enhancement bonus to the attack rolls and the damage rolls of warlock powers, as well as warlock paragon path powers, that have the implement keyword. Without a rod or a wand, a warlock can still use these powers, but he
or she doesn’t gain the bonus provided by the magic
implement."
PHB page 131
(This is just one out of several similar passages, one for each class)
Do note that warlock implements specifically work only for
1) warlocks
2) warlock powers
And also note that the rules address the issue when you don't have the proper implement - you can still use your powers, but you don't gain anything implement-related. (See my previous post)
So a Paladin with a Warlock power is in the exact same situation as a Warlock having lost his rod or wand.
The fact the Paladin have a Holy Symbol is in no way relevant to getting any implement-related bonuses.
And you won't see it written anywhere you need this or that, because in exception based design only the exceptions are called out.
And as your implement isn't a rod or a wand your Holy Symbol can't be used to give implement-related benefits for your Warlock powers.
This however
*IS* explicitly spelled out:
"In addition, you can use a rod, a wand, or a pact blade as an implement when using a warlock power or a warlock paragon path power."
PHB page 208
Note how the multiclass feat explicitly spells out that you need a warlock's implement for your warlock powers.
As mentioned upthread, there is a way to get around this: if you choose Corellon as your patron deity, you can use a Star of Corellon, which allows you to use that holy symbol for "any arcane" powers, which obviously includes Student of Caiphon.
Do note that this is supposed to have a real cost in the game: not being able to use any other holy symbol.
So the RAW is clear:
In general, the game assumes you use a second (presumably inferior) implement for your multiclass implement powers (and copes with the inconvenience of having to keep a hand free for this purpose).
Or you pick up the specialized implement for your class that can be used for the powers you've gained by multiclassing, and thus abstains from getting any other powers and properties for your holy symbol.
Feel free to houserule this, but I agree neither to any "this is how the rules can be interpreted" claims nor to the "there isn't any balance or power issue" evaluation.
WotC clearly views flexibility as a power boost in itself, and multiclassing thus carries a real cost in lowered effectiveness to compensate. This cost is not trivial, so by houseruling you need to be aware you are removing a non-trivial cost.