I don't recommend a generator; it's more equipment, more weight, more fuel, and noisy.
Let's see how to solve your problem. First of all you need to measure and understand what is happening.
1. You'll need a battery system monitor or some gauges so you can measure battery voltage, house amp usage, and alternator output. (Even without this existing problem you will find the information from these extremely useful). Maybe you already have a way to do this.
2. Measure the house usage. If the usage is 10 amps, then you use 240 amp/hour per day. You'll need measure the amp usage for each device, then calculate how many hours per day you use them, and add them all up. My boat uses about 150 amp/hours. (A key usage will be your refrigerator; this may be reduced by adding insulation or turning it off at night, or setting to a higher temperature).
3. Look at your house battery capacity. You can reasonably expect to use about 50% before the voltage gets too low (below 11.5). If you have two 165amp 12v batteries you'll get a total house amp/hour capacity of 330 amp/hours but you can only use about 165 amp hours.
4. Each day, when you use up the available battery, you'll have to recharge. Often boats will have difficulty going over 24 hours without re-charging. Or you need to add more batteries or reduce the daily load. Somehow you need to achieve a balance.
5. Measure the amp output , actually coming out of the alternator, and watch the pattern over time. With a good regulator your alternator should start at close to its rated amps and then run at 80% of its rated amps for an hour or two. You can calculate how many hours it takes to recharge your house battery bank.
What to other people do?
We charge our batteries with the main engine every day, and have done so for many, many years.
Our boat has 450 amp capacity house battery. We can go about 36 hours on a full charge. On a long weekend we need some help. We have two big solar panels which can be aimed to the sun and are usually shade free. We get about 60-80 amp/hours a day from the pair. With our daily usage of 150A/h we need to get another 90amps a day from the main engine (we have no gen-set). We have a 130amp alternator and a smart regulator. With this we need about 1 hour engine running per day. We do it just before cocktail time each day when the solar has tapered off.
When sailing we may need to run the engine longer because of the heavy electronics usage (chart Plotter, etc)
In cold climates we can run the engine less, often only every other day, because the cold water reduces the refer load.
Even if the batteries are low in the morning when we get underway the electronics will come up as soon as the engine is running, so if they don't for you, this is a charging or battery problem.
