If I was experiencing pain in the bench press, I would aim to fix the problem, while only temporarily employing alternative exercises for chest hypertrophy.
Firstly, to answer your question, alternative exercises for chest hypertrophy would include chest flies (with dumbbells or cables), plus the push-ups and dips that you've already mentioned. However if you can do 30+ repetitions of pushups, then you would likely benefit from a more difficult, and ideally externally loadable, exercise choice.
However I wouldn't just abandon the bench press and seek to replace it with one of the above. I'd suggest reading up on strategies for managing pain in the gym, such as those published by Barbell Medicine (video, article), in which you would aim to find a way to complete the exercise without pain, and work back towards normality from there. Procedurally, this would look like:
- Try to find a lighter load that you can tolerate without pain.
- If the pain occurs at all loads, then try to find a reduced range of motion that you can perform without pain.
- If the pain occurs through all ranges of motion at all loads, only then consider alternate exercises to substitute. This would include firstly variations on the exercise that produces the pain, such as experimenting with flat-back vs arched-back benching, moving to an incline press, or failing those then moving to dips or pushups.
- Gradually attempt to work back towards being able to bench press at challenging loads. Reassess if the pain comes back.
If you know that you can perform pushups or dips without pain, then I'd be included to hugely reduce your load on the bench press, add in some pushups or dips to preserve strength, and then spend a month or two working back up to your original bench press weights, while gradually removing the dips as bench press became more challenging. I'd also ensure I was doing some direct work on the area where the pain occurred, such as one-arm DB rows, if it could be done without pain.
E.g. If you were benching 60kg for 10 reps for 3 sets, twice per week, and felt pain with any load heavier than 50kg, then an 8-week recovery program might look like this:
- Bench press 40kg 2x10, 1 set dips (gradually introducing the dips to give your shoulders a break and avoid DOMS from a sudden introduction of a new exercise)
- Bench press 42.5kg 2x10, 2 sets dips
- Bench press 45kg 2x10, 3 sets dips
- Bench press 47.5kg 2x10, 3 sets dips
- Bench press 50kg 3x10, 2 sets dips
- Bench press 52.5kg 3x10, 2 sets dips
- Bench press 55kg 3x10, 1 set dips
- Bench press 57.5kg 3x10
- Bench press 60kg 3x10