As chirlu says, the usual way to form a feminine equivalent to a noun in -(s)sor is with -strix. Aside from the examples mentioned in that answer, here are further words of this form found using Logeion's reverse search tool:
There isn't really a neuter version of this suffix; there are several suffixes, some related and some unrelated, used to derive neuter instrument nouns
The commentary on Virgil's Aeneid by Servius (or Pseudo-Servius?), when discussing the form victricia, explicitly states that masculine -tor and feminine -trix have no neuter counterpart in the singular, and that the neuter version in the plural ends in -icia (used adjectivally in Virgil):
victricia omnia nomina a verbo venientia cum in 'or' exeunt, masculina sunt, ut victor; cum in 'trix', feminina sunt, ut victrix; neutra vero non faciunt, nisi tantum ex numero plurali: unde est victricia.
The suffix -trum
There is a neuter suffix -trum, but unlike -trīx, it does not form nouns that are synonymous aside from gender with masculine nouns in -tor. The suffixes -tor and -trīx both form agent nouns. But neuter words in Latin almost always refer to inanimate things rather than beings with agency, and so in the words where -trum appears, rather than denoting an agent, it instead denotes a tool or instrument. We see this in one of the few cases of a pair of -tor and -trum words, arātor "ploughman" and arātrum "plough".
However, -trum was never a very productive instrument noun suffix in Latin. One reason might be that it appears to have originally occurred only in specific phonological contexts: when there was an /r/ or /l/ sound earlier in the word, or when there was an immediately preceding /s/ sound (and actually, I think even the -strum words more often than not have a preceding /r/ or /l/ sound; not sure if that's a coincidence). For more on this restricted pattern, which seems to be old, see the question Suffixes -τρον, -θρον, and -εθρον. Profess- technically meets both of these criteria, if you include the prefix, so profestrum does seem at least supportable by analogy (moreso in fact than the already-coined neologism "computātrum"), although it's unclear what an instrument or tool noun corresponding to "professor" would be.
Neuter instrument suffixes other than -trum: -culum/-crum, -tōrium, -mentum
In general, the suffix -culum (ultimately from the same etymological source as -trum) appeared on a greater number of Latin instrument nouns than -trum (although neither suffix was as productive as -tor ended up being). When preceded in the same word by -l-, -culum frequently undergoes dissimilation and becomes -crum.
Usually, -culum attaches to the present stem of the verb (possibly with changes in the preceding vowel), rather than being built on the perfect participle stem. So for example, from offendo, -di, -sum there is offendiculum. An example derived from a second conjugation verb is irrīdĭcŭlum. This suggests that *profiticulum should be possible, at least in form; however, I don't like the look or sound of it.
In later Latin (especially), the compound suffix -tōrium becomes somewhat common and productive for forming nouns for tools or instruments, such as cōnflātōrium "furnace" and pūnctōrium "instrument for pricking". From profiteri, we would expect *professōrium.
The suffix -mentum also comes to be used sometimes in later Latin to form words for tools or instruments, such as involūmentum in Augustine for classical involūcrum "wrapper, covering, envelope". I believe adding this suffix to profiteri would most likely yield *profitumentum (like documentum, monumentum; this vowel is what etymologists call the "sonus medius" and so a variant in -imentum might also be possible).
Overall, I would say there is no systematic parallelism between -tor and any neuter ending the way there is between -tor and -trīx.