[RPG] What Hit Dice do multiclass characters spend at the end of a short rest

The relevant quote is this on page 15 of the PHB, or here in the basic rules:

When your character gains a level, his or her class often grants additional features, as detailed in the class description.

Nowhere does it mention that it takes any time to gain these features when you level. The only exception is that it explicitly states that leveling increases the hit point maximum. Since 5e defines current hit points and maximum hit points as specific terms, you are right to conclude that the you don't gain your new hit points until you rest. However 5e doesn't make that distinction in the terms of hit dice. Like your proficiency bonus, attribute gains, etc., it takes effect immediately upon leveling.

It is reinforced by the fact that Hit Dice advancement is in its own sentence:

Each time you gain a level, you gain 1 additional Hit Die.

Then the following sentence states:

Roll that Hit Die, add your Constitution modifier to the roll, and add the total to your hit point maximum.

Which make it a separate point.

Before all considerations, you've got to ask yourself what is this particular optional rule trying to achieve.

In the text block preceding the one you quoted, you can find that goal:

Healing:

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These optional rules make it easier or harder for adventurers to recover from injury, either increasing or reducing the amount of time your players can spend adventuring before rest is required. (DMG, p. 266)

By adding Healing Surges in combat, the DM aims to make hit points easier to regain between long rests.

Thus, allowing the expense of hit dice both in combats and short rests seems to concur to that goal.

As for the order in which things occur at the end of a short rest, it's up to the DM's decision (see it as fine tuning the difficuly of the game). Having the players use any hit dice left before regaining 1/4 of their maximum number will ensure they will have at least one to spend on the next encounter. On the other hand, allowing the 1/4 recovery before any short rest expense of hit dice will lead to healthier PCs at the start of the next fight.

tl;dr : Yes, you can use both. On short rests, the DM decides if you regain some hit dice before you expend any, or the other way around.