Togekiss is one of the most threatening Pokemon in the game. Dynamax is also worth a lot, so it's really unreasonable for you to expect to win a 1v1 without spending your own Dynamax unless you have a dedicated counter or is willing to ignore niche options, or possibly both.
Looking down the VR / Usage:
Mimikyu is a generally strong 1-time check to Togekiss. It is faster, so they can't really afford setting up a Nasty Plot, and if they don't then you can just Dynamax and 2HKO with Max Phantasm, and if they are bulky, you can finish off with a Shadow Sneak with its decreased defenses.
If you want to spend the Dynamax on something else, Mimikyu also has access to Trick Room. It can set it up, and sacrifice itself. Then bunch of Pokemon are in a great position to KO it with its Speed boosts working against it.
Weakness Policy Max Rockfall Excadrill is a really good check because it generally beats the Togekiss 1v1, activates its speed boost and Weakness Policy. Obviously it involves matching Dynamax for Dynamax. Focus Sash Excadrill is also a fine set, but that generally means that you want to at least put some prior damage on the Togekiss so that you can KO it.
Weakness Policy Tyranitar wins 1v1. However, it cannot switch into Max Starfall. However, if it were to get the matchup without losing much health, then Tyranitar can tank even a +2 Max Starfall (from Weakness Policy or Nasty Plot), activate its own Weakness Policy, and OHKO Togekiss.
Rotom-H has the perfect defensive typing to wall Togekiss's STAB + Fire move coverage. It does not need to Dynamax to beat it, as Assault Vest + Discharge is a quick stop to Togekiss, but Ancient Power sets will get around it.
Darmanitan-G is an offensive check. Banded Icicle Crash will OHKO the Dynamax entirely, and Scarf still outspeeds the +1 Togekiss and does 75% - which is a ton of damage.
AV Snorlax is pretty much a hard stop to Togekiss. It actually also is able to drop Togekiss's speed with Max Strike, too, which is actually really nice. It means that even if Snorlax goes down after Dynamax ends due to an unfortunate string of Flinches, it will be enough to neuter it.
Sash Cloyster and Aegislash can take a hit and do a lot of damage, but obviously it's not a full-on check, the same reason why your Excadrill isnt a foolproof check.
Ditto can Dynamax and copy its boosts, Durant can one-shot an unboosted Dynamax Togekiss, Gastrodon can Mirror Coat or Yawn it as it takes minimal damage.
There's almost no safe way to hard-switch a Pokemon straight into Togekiss, not use its Dynamax, and survive the encounter and beat it 1v1. That's just being greedy.