An alleged Trump "___" to Epstein was sexually suggestive.

The choice that makes the most sense in the context of the statement about an alleged Trump connection to Epstein being sexually suggestive is a birthday note. A birthday note may imply a personal and informal communication, which could feasibly contain content that is more suggestive in nature. In contrast, an invitation typically denotes a formal request to attend an event, which is unlikely to contain suggestive content.

Similarly, an announcement is generally more formal and straightforward, lacking the personal touch that might lend itself to suggestive undertones. A letter of intent is typically a business-related document and would not be an appropriate format for something that includes sexual suggestions. Therefore, a birthday note best fits the description provided in the question, allowing for a scenario where intimate or suggestive language might appear.