It might be, maybe you‘ll correct this assumption, that the push-up style is too hard on your triceps. Do you find the "hand under the shoulders" to be awkward and you move your hands out to put less strain on your triceps? Moving your hands out balances the stress from the triceps to the shoulders and outer chest area.
Unfortunately, the CF likes the tricep style it seems.
By keep practicing the tricep or CF style of push-up, you‘ll improve slightly over time. What might help is going to the gym and performing shoulder and chest exercises. Doing the bench press, inclined bench press, the shoulder press, lateral side shoulder raise, and the lat pull down builds the muscles in the push-up area. You may not want to over-train the tricep too much and risk a tendon pull at the elbow.
Try to find a weight that you can perform all the exercises above 8 to 11 repetitions. Find another weight that you can do the same exercises 4 - 7 repetitions. The less repetitions the more weight.
As a suggestion, do the exercises for one week at 8 - 11 reps and start at 3 sets each. Next week do 4 - 7 reps at 4 sets each. Next week after do the exercises at 8- 11 reps at 5 sets each. Next two weeks do the same with 4 - 7 reps at 5 sets.
Go back to week one and keep the cycle. You may want to do this about 3 to 4 times a week.
Likely you‘ll be using more weight each cycle. Then try the push-ups. You‘ll find you‘ll do them faster with more repetitions. It may take two cycles to see a significant improvement and it takes time and patience.
These weight exercises could be part of an overall fitness program and I‘m just offering it as an example. There are weight lifting programs designed for a full body workout that work the legs, back, abdomin (sp), mid-upper back, shoulders, chest, biceps and triceps. All muscle groups need focus and I‘d suspect will be used during BMQ lifting, carrying, running, and PT.