DPS CALCULATOR (GO TO LIZZIE TAB -> MAKE A COPY TO USE) : https://docs.google.com/spreadsheets/d/1B6djrTU7aQJntaAMRajOiBlR7F-L35DiGbg0WhIedmI/edit?usp=sharing

This builds looks weird at first glance and may raise many questions:

Where is the heat damage?
Temple's 3rd ability Ripper's Wail applies heat damage as a separate non-combining mod (similar to [Shock Trooper] and other augments) and boosts heat DoT but additively to other +heat mods like [Thermite Rounds] and [Hellfire].

Why there is no modded critical damage?
The critical damage comes from Primary Blight which gives 144%, even higher than [Vital Sense]. This stacks almost instantly from the toxin procs on [Lizzie].

Critical chance comes from the combination of Arcane Hot Shot on the Warframe (even just r1 gives 100% cc while r5 gives 300% cc modded), [Critical Delay] (200%) and flat critical chance from Overdrive (around 100 to 200% flat depending on strength). The final CC we end up with should be over 300% which is guaranteed red crits.

Why Blast + Toxin?
Toxin is modded to activate Primary Blight which gives both multishot and critical damage. Multishot is VERY important on beam weapons like [Lizzie] because multishot DOUBLE DIPS status DPS on continuous weapons. Blast is modded into order to allow Toxin to not combine with innate heat to make Gas, and Blast itself can be a useful trash clearing status that helps kill densely packed enemies. As a final bonus, [Archon Continuity] modded on the warframe causes toxin procs from [Lizzie] to simultaneously proc Corrosive status which contributes to both armor strip and [Galvanized Aptitude].

Why not viral?
If you aim down sight on [Lizzie] you can simultaneously shoot the priming waves and primary flamethrower without interruption. For general use you will want to just spam Viral waves (aim and tap 1) everywhere to boost all damage to health up to 4.25x at 10 stacks. You could also subsume on Nourish for free viral without needing to use alt-fire.
Against eximus and thrax you can tap 2 for magnetic which boosts damage to overguard and shields
Since you have corrosive status coming from [Archon Continuity] already, this means you don't need to mash so many keys at once and can focus on just priming viral.
Cold priming can be somewhat helpful in giving flat CD boost of 1x at 10 stacks but this is not mandatory. Cold may also be useful to build stacks of Primary [Frostbite], in case you don't own any Primary Blight. This means you just need to spam both viral and cold into enemies.

MOD BREAKDOWN

• Galvanized Chamber: mandatory multishot, which double dips beam status DPS
• [Galvanized Aptitude]: boosts status chance and also gives +80% damage per status, additive to strength on [Temple]. With 300% additive equivalent damage, [Galvanized Aptitude] with Blast Toxin Heat Corrosive Viral (5 statuses) provides a further +400% damage, which is a relative 2x multiplier on top of the existing strength multiplier.
• Critical Delay: gives critical chance
• [Thermite Rounds] + [Rime Rounds]: Makes blast and adds status chance. [Primed Cryo Rounds] would give more raw damage but Ripper's Wail already gives way too much additive elemental damage, and this comes at the cost of status chance which is more important.
• Malignant Force: Makes toxin and adds more status chance
• Primed Bane: multiplicative to damage, double dips status against a specific faction. Flexible with Rifle Elementalist if you don't like faction mods
• [Vile Acceleration]: Gives firerate which directly multiplies damage by 1.9x. This can be flexible if you offload with external firerate boosts (e.g. abilities, arcane acceleration, reinforced bond)

FLEXIBLE OPTIONS

• Rifle Elementalist: multiplicative 1.9x multiplier to status DPS
• Vital Sense/Hammer Shot: add even more critical damage, hammer shot also provides status chance
• Magnetic Capacity: gives magnetic without needing to shoot the alt fire, could be good quality of life for pure thrax killing
• Vigilante Armaments: gives more multishot which double dips status DPS
• Serration/[Amalgam Serration]: more additive damage which is overall a pretty small boost

[Lizzie] QUIRKS
[Lizzie] seems to be almost exactly like [Ignis Wraith]. It is a flame thrower beam weapon with a 10m range that can be boosted by [Sinister Reach] to have a final range of 22m. At the end of the beam there is 4m spherical AOE that like all hitscan AOE will IGNORE multishot entirely. This effectively means the actual final [Reach] of the beam is 26m but the enemies at the final 4m end will only receive about half to a third damage, while the enemies between 18 to 22m will actually receive about 1.5x more damage.