Help not failing Machamp solo

Shadow Ball should be changed on your Alakazam for Futuresight.

Your first two listed are fine, but the rest of your team are way too low level, or not optimized.

Latias does not have the DPS to compare with optimal counters, and yours is very low level. At the very least, it’s an anchor coming in after the Pokémon that get you way ahead of the clock.

Exeggutor should at least carry Solar Beam, but even then yours is too low level and vastly outperformed (if you have one with Confusion that works better too). Exeggutor should overall not be used for Machamp solos. It’s pretty borderline as well.

You should work on building a team of strong psychic types (Espeon, Alakazam, Gardevoir etc)

The Dragonite is okay as an anchor.

Couple of thoughts for you. Your Espeon should be FTM to confusion. I personally use Exeggutors to round out my Champ team for the bulk as I do not have Mewtwo but they are both E/P. Solar beam is usable but seed bomb is not. I also agree with the previous response about the SB Zam not being good enough. That really sucks about your lack of luck during the Kanto event. Just as an example, my current Champ team is..
2746 Alakazam C/FS
2815 Espeon C/FS
2800 Espeon C/FS
2684 Alazazam PC/P
3318 Dragonite DT/H
2543 Exeggutor E/P

Hope this helps a little.

TM Alakazam to Future Sight.

TM Exeggutor to Psychic or Solar Beam.

Just transfer that Latias, too low attack, and replace with something else like Gengar/Slowbro.

Espeon, how bad is it? Confusion is slightly better generally, though.

Use Mewtwo. Even shadow ball Mewtwo puts you ahead the timer against Machamp. If you have yet to put together your 6-man team of Machamps, it's worth the rare candies (and shadow ball Mewtwo is a great generalist). Lugia is better as anchor than either Latias or Dragonite even when raiding against heavy slam, and you probably have a lv 20 one at least too? Gardevoir will help a lot against dynamicpunch (the most difficult one). The seedbomb and shadowball mons could be switched out for more dragonites if you have them, they really need super effective attacks to be of any use. Yeah, and your Latias has outrage, that too. Latias with double psychic moveset performs worse than exeggutor with extrasensory/psychic on the timer department (even extrasensory/solarbeam, though it is bulkier).

Given your current levels, you may also want to be strategic when you attempt to solo a Machamp, i.e. try during windy conditions when your psychic pokemon are weather boosted.

In my personal strategy I always combine a glass cannon with a bulky "viable" option. So in your case, I would make this order:
Espeon
Dragonite
Alakazam PC/FS
Exeggutor
Alakazam PC/SB
Latias

This allows to absorb the damage better when Machamp fires off 2 consecutive charge moves, so that it doesn't cripple immediately those glass cannons that win the clock for you.

About the Espeon, I think it's more complicated than what others have said here. I generally prefer Zen Headbutt because of dodging, but you obviously have a DPS issue... on the other hand, if efficient dodging allows you to fire off 1 or even 2 more Future Sights, then it might still be a better choice for you. Anyway, if your current buddy Eevee is high level and you get Future Sight, it will save you a lot of trouble changing it for Latias.

In any case, I think your best chances are soloing easier Tier 2 or 3 bosses to get more TM's and good weather boosted mons. For example, your Exeggutor will be fantastic with ANY of the other moves, and you can still get weather boosted Eggies from Tier 2 that may have the moves you need (a bit of stardust investment would be needed anyway). Gengar is also great with triple resistance to fighting and STAB Shadow Ball, and it's for me an easier Tier 3 boss. Also, you can try to locate an Ex Raid eligible gym and do all lower-tier raids. Not only you will get a Mewtwo (best DPS against Machamp), but it will also help you to get in contact with other local players and ask them to join their community or start one. Raiding with other trainers makes this game so much easier and fun that it's worth it.

If you get 2 more Alakazam/Espeon to replace your Latias and Exeggutor your team would be better than the one I used to solo all movesets for a few months. Lots of dodging required against DP and HS but doable. Alternatively TMing the Exeggutor to SoB or Psychic and powering up the Latias also works but that costs premium items.

Machamp seems to be a permanent raid fixture so you should have time to optimize your team.

Definitely FS that 3rd Zam, hunt down all the Exeggcutes you can especially weather boosted ones. Even a boosted Xatu with AS/FS would be better than that Latias

Also in case you're not aware, you can bag more Espeons by walking Eevee 10km and evolving him from the buddy screen in the daytime.

If you're fainting long before the timer runs out then you're either not dodging enough or not when it's important to dodge e.g. to get off another charge move before fainting, or your Pokemon are underlevelled.

If you're timing out, you don't have enough DPS, which can be because of suboptimal movesets, you're dodging too much, or again, your Pokemon are underlevelled.

Besides the Latias I think your Pokemon are at high enough levels.

Alakazam with PC + SB is too slow but TMing either move is acceptable. PC + FS is better than C + SB but maybe you don't have many charge TMs, Exeggutor with Seed Bomb also needs TMing but either Psychic or Solar Beam is viable.

Latias is underlevelled - it's by far the weakest link on your team, a single Latias will put you behind by about 9 seconds, but even at level 30 with the optimal moveset (ZH + P) it just about clocks in at 180s - so there's plenty of other Pokemon that could replace it, for example another Dragonite with DT + O, doesn't even need Hurricane.

If you're still struggling after getting optimal movesets and upgrading / replacing the Latias, you can power your Pokemon to their next breakpoints, that extra bit of damage on each of your Pokemon's quick moves might make the difference.

You could power the Exeggutor to at least level 29.5, because Extrasensory will go up from 14 to 15 damage, for 23k dust and 20 candies.

The level 32 Alakazam can be powered to 34 where Psycho Cut will go up from 7 to 8 damage, for 26k dust and 28 candies, a slightly better investment than the Exeggutor since Psycho Cut is almost twice as fast, but both are your lowest cost options.

Seriously Mewtwo. The guy has 4 Latias. I know a lot of people are salty they can't get invites, but at least 1 of his raid buddies should have a Mewtwo. Raid twice to thrice a week where he caught it and in a less than a month he'll have the first invite ready. Alakazam w shadow ball can be used on gengar solos ie, no need to change that for just one raid.