かねる

1. 1Cannot do something for reasons of position or circumstance; used as a polite, softened refusal of a question or request.Ex. 1
2. 2Politely declines to comply with a request while acknowledging the other party's situation.Ex. 2
3. 3Finds it hard to bring oneself to do something, or hesitates over it (e.g. 決めかねる 'unable to decide', 見るに見かねる 'unable to stand by and watch').

Typical of formal business speech, customer service and official notices, where the speaker declines a request without saying a blunt 'no'. In this refusal use it usually describes the speaker's own side and implies the refusal is due to rules or circumstances rather than personal unwillingness. Unlike わけにはいかない, which states a social or moral constraint in more everyday language, かねる sounds more formal and deferential.

• かねないN2「かねない」 looks negative but actually means 'there's a strong chance something bad will happen,' expressing worry or warning; 「かねる」, by contrast, means it is hard or inappropriate to do something—the two meanings are nearly opposite and cannot be swapped. Both attach to the verb stem (masu-stem).
• がたいN3「がたい」 also attaches to the masu-stem and conjugates like an i-adjective. It expresses psychological difficulty accepting something (as in 「信じがたい」) or objective near-impossibility (as in 「得がたい」), emphasizing that something can hardly be done. Its tone is formal/written; when politely declining customers, use 「かねる」 instead.
• にくいN4「にくい」 describes difficulty caused by the nature of a thing or objective conditions, including phenomena that don't easily occur (such as 「壊れにくい」); its tone is neutral. 「かねる」, on the other hand, refers to being unable to act due to one's position, propriety, or hesitation—it is more polite/formal—and generally they cannot replace each other.
• ようがない / ようもないN2「ようがない」 indicates there objectively is no means or method, so you simply cannot even start; its register is neutral, and 「言いようのない」 can express extreme degree. 「かねる」 means one finds it awkward or improper to do something because of their stance or feelings, so the two cannot be interchanged.
• わけにはいかないN3Both convey reluctance/inability to do something out of propriety. However, 「わけにはいかない」 attaches to the dictionary form, has a neutral register, emphasizes self-restraint based on reason, and with the nai-form also means 'must do'; 「かねる」 attaches to the masu-stem, sounds more formal and deferential, and suits politely refusing customers.
• 得ないN2「得ない」 likewise attaches to the masu-stem, but it means logically impossible or beyond one's ability/conditions—its tone is blunt and forceful, with none of the softening nuance needed for polite refusal. Pure inability (lack of skill/capacity) cannot take 「かねる」, so the two generally cannot be interchanged.

かねない (despite looking like the negative of かねる) means 'might well happen', and がたい and にくい express psychological or objective difficulty (信じがたい, 読みにくい) rather than a polite refusal. ようがない means there is no possible means of doing something, わけにはいかない gives a social or moral reason in everyday speech, and 得ない is a formal way of saying something cannot be done or cannot be the case (予測し得ない 'cannot predict', ありえない 'impossible') without the courteous-refusal nuance of かねる.