There is in this sentence a basic locution, "à corps perdu", which you can combine with picked verbs, here "lancer". We get the literal signification from the TLFi below.

Loc. adv. À corps perdu. De toutes ses forces, avec une fougue que ne tempère pas la perspective de la fatigue, la crainte du danger
ex. : Se jeter, s’élancer à corps perdu dans la mêlée. Fatigué de s’en être tant donné, car les enfants font tout à corps perdu, il s’endormit

You see from the definition that some of the picked verbs can be "lancer" and "s’élancer"; there are others verbs and other possibilities of combination : dévaler (une pente,…), ; instead of using the locution as an adverbial modifier you can use it as an adjectival one : une poursuite à corps perdu,…

There is a figurative sense to this locution, to be found also in the TLFi.

Fig. Avec passion, sans hésitation et sans mesure, sans retenue et sans prudence. Je me lance à corps perdu dans les lectures religieuses (Flaub., Corresp.,1846, p. 415).

In your sentence, this is of course the relevant sense. The verb used with it takes on also a figurative meaning. A translation of "lancé à corps perdu" would be (DeepL) "thrown with full body", (Harraps) thrown recklessly, thrown without restraint.

À corps perdu is an expression that indeed literaly translates to "at lost body".

It actually means something done without caring about damages that may affect the involved body, up to full lost.

There is at least another similar expression: à fonds perdus, that means "done without caring about the induced cost / whatever the cost / knowing all money thrown to the project will be lost".

Lancé means "in movement after having been launched". Note that the expression is lancé dans une course poursuite, i.e. more or less "running in a chasing race".

I might then translate that phrase that way:

The republican candidate, engaged in a desperate chase against Joe Biden…