Weightlifting is most often a noun, and nouns don't have past participles. The corresponding verb is usually expressed not by a single word, but by the phrase "to lift weights". I haven't found any entries in a traditional dictionary for a verb weightlift, although you can see it, marked "rare", in Wiktionary.
The incorporation of an object into a word is most commonly seen in nouns or adjectives derived from verbs, not so often in verbs themselves, as I mentioned previously in relation to the adjectives spellbound and spellbinding and the noun blameshifting.
There is no editorial guideline that says to use a hyphen only in some parts of a verb, but not in other parts of the same verb. If you for some reason decide to use "to weightlift" as a verb, its past participle would be weightlifted and its gerund/present participle would be weightlifting. If you use "to weight-lift" as a verb, its past participle would be weight-lifted and its gerund/present participle would be weight-lifting.
There aren't general rules about whether compound verbs like this, or compound words in general, should be spelled with or without a hyphen. But the general trend in recent centuries has been towards less use of hyphens than in the past: "weight-lifting" itself used to be a commonly used spelling, but has now been overtaken by weightlifting. I'd therefore advising using the non-hyphenated spelling if you make use of a verb weightlift, weightlifted, etc.