Technically,

uniquement = one of a kind
seulement = only
exclusivement = exclusively

They are all sometimes interchangeable, but not always.
For example, you cannot say

Il reste uniquement/exclusivement 3 pommes.

These three adverbs are considered synonymous. I have mixed feelings about the slight nuance between uniquement and seulement expressed in your question.

While I would probably prefer the first sentence too, I have no real issue with:

Je te demande donc de nous servir de guide, mais seulement de guide, et uniquement jusqu’au lac.

About exclusivement, I would say exclusivement de guide is conveying the same idea that uniquement de guide but perhaps with a slightly stronger way. On the opposite, simplement de guide would be softer than uniquement.

I wouldn’t use exclusivement jusqu’au lac. This sounds odd to me. Perhaps the reason is the other tracks are not in frontal competition with the one to the lake while with exclusivement de guide, other roles might be played at the same time by the person the sentence is directed to. I wouldn’t use simplement jusqu’au lac at all here. Simplement might be used that way: je veux simplement aller jusqu’au lac meaning I have no other plan than going to the lake.

Definition

Uniquement, seulement and exclusivement share their meaning, moreover when talking – aka when words don’t hold all the meaning.

uniquement wiktionary meanings :

• Exclusivement à tout autre.
• D’une façon unique, au-dessus de tout, préférablement à tout.
• D’une façon excellente.

seulement wiktionary meanings :

• Uniquement ; rien que ; sans quelqu’un ou quelque chose de plus.
• À l’instant.
• Du moins ; au moins.
• Toutefois ; mais.

exclusivement wiktionary meanings :

• En excluant.
• Uniquement.
• En ne comprenant pas.
• À l’exclusion de toute autre chose.

If you are unique, you are alone in what you are, having a trait exclusively.
If you are alone, you are, in a particular environment, the only one, so unique and exclusive (no sharing).
If you are exclusive, you are not sharing, so unique of-a-kind and therefore alone in some way.

Example

[original] : Je te demande donc de nous servir de guide, mais uniquement de guide, et seulement jusqu’au lac.

vs

Je te demande donc de nous servir de guide, mais seulement de guide, et uniquement jusqu’au lac.

vs

Je te demande donc de nous servir de guide, mais exclusivement de guide, et uniquement jusqu’au lac.

Using something else than seulement sounds like you are underlying the more important part of your sentence, since you are using a particular word. By using exclusivement + uniquement sounds like both are important but for different reasons.

Conclusion

Basically, when talking, they may be used according to your pleasure.

But if you want to be idiomatic, and don’t shock natural speaker you have to use it according to what you want to express :

• uniqueness
• loneliness
• exclusivity