I believe that this is actually a case of the genitive of quality - describing an attribute that would ordinarily be described by an adjective (see, e.g., Blass et al., A Greek Grammar of the New Testament and Other Early Christian Literature, §165). I think this conforms to your definition of the "subjective" genitive. Another example is Luke 16:9:
And I say unto you, Make to yourselves friends of the mammon of unrighteousness; that, when ye fail, they may receive you into everlasting habitations.
Κἀγὼ ὑμῖν λέγω· ποιήσατε ἑαυτοῖς φίλους ἐκ τοῦ μαμωνᾶ τῆς ἀδικίας, ἵνα, ὅταν ἐκλίπητε, δέξωνται ὑμᾶς εἰς τὰς αἰωνίους σκηνάς.
In the case of 1 Timothy 3:6, the meaning of ἵνα μὴ κρίμα ἐμπέσῃ τοῦ διαβόλου would be something like "lest he fall into the sort of condemnation that the devil fell into."
This seems to be borne out in how the Greek Church Fathers understood the verse. John Chrysostom, a 4th century Byzantine Greek, for example, comments:
Do not immediately advance to a station of dignity a novice, that is, one of these new converts. For, if before he had well been a disciple, he should at once be made a Teacher, he would be lifted up into insolence. If before he had learned to be under rule, he should be appointed one of the rulers, he would be puffed up: therefore he adds, Lest being lifted up with pride, he fall into the condemnation of the devil, that is, into the same condemnation which Satan incurred by his pride.