、、。
They were inviting me last week too, but because it ran late, I turned them down tearfully.
Literal
Last-week around [also-も] was-inviting [for-me-くれて] [but-が], night [subject-が] late [because-ので], tearfully turned-down.
A sentence with a nice adverbial kicker. 泣く泣く literally 'crying crying,' is a reduplicated adverbial form used to mean 'tearfully,' 'reluctantly,' 'with deep regret' — often idiomatic for making a painful choice against one's heart. Not literal crying, but emotional reluctance. This reduplication-as-adverb pattern is productive in Japanese: 泣く泣く, 嫌々 (いやいや, reluctantly), 渋々 (しぶしぶ, grudgingly). 誘ってくれていた chains ~てくれる (favor from someone) with ~ている (ongoing/past state) — 'was inviting (me, as a favor).' The ~あたり attached to a time expression marks approximation: 先週あたり = 'around last week.'