I was assuming the 1000 A battery was just CCA rating for a starter battery for cars , in other words the Cold Cranking Amp rating.
These are not suitable for many long cycles of charge discharge due to the thin plate design , but are cheap and dirty.
The weight of a dry battery will tell you something about its plate lead content.
For a 135 W charger @ 14.2V max or 10A, and 14.2 being 77% of the 18.5 Voc, it is already well matched for MPT. Cloudy day Voc will drop to 17.5V then this ratio of 81% is still well within MPT range.
Generally chargers are CV limiters like alternators in cars set to 14.2V since the 1000 CCA rating far exceeds the 10A capacity of source.
You don't want to discharge your battery less than 11.5V for low current load voltage but startup surge from the SS inverter is allowed drag it down ( 7V for 1000A or much lower A if discharged or 0% SoC)
These are the general rules around a good battery charger, in addition to protection for reverse voltage, shorts and load dumps.
When the charger is switched off the float voltage will drop 1.7V so you need hysteresis in sensing if you choose a cheap relay On-OFF control method.
A linear method at 10A and 2.5V drop will not work well as it dissipates 25W.
The better approach used in regulators is a hysteretic PWM voltage regulator or a SMPS.
You can buy a board off EBAY for $10~$30 or roll your own for giggles and kicks for $10 if you know how. More if you don't.
https://www.ebay.ca/itm/10A-Solar-C...619?pt=LH_DefaultDomain_0&hash=item2c99a41223
https://www.ebay.ca/itm/10A-12V-24V...253?pt=LH_DefaultDomain_0&hash=item1c46aae565
Note that these cheap regulators don't use any magnetics or MPPT, so the current will be pulsed, EMI may be abundant in the AM band but it works hey for $8 ~$12 + shipping and has digital features.